arXiv:1503.04365v1 [physics.chem-ph] 15 Mar 2015 he ocpsaeimportant. are concepts three 18. force Ref. atomic see experiment, mode water, mea- tapping de-aerated with the microscopy. in obtained images attraction of from surfaces, reduced features hydrophobic a various between ing forces from sured confirmed been ihdi 1994, in lished nriso h oi iie ytelqi-ao surface liquid-vapor surface the in by difference divided cosine the tension. solid to the the equal that of is says energies angle which contact sur- the equation, of to Young adhered the case bubbles twice the faces, and to for nucleation Third, equal heterogenous radius. amount of by an divided pressure by tension the surface liquid than the larger surrounding is the inter- bubble the the of of that pressure says gas nal which equation, Second, Laplace-Young to- vanish. the the energy) of free derivatives the the (equivalently which entropy at tal point the is which dius, inwt h xeietlyosre tblt (hours stability observed experimentally the with tion models. physical realistic thermody- conventional and must with principles nanobubbles namic points of four existence these the reconcile for explanation Any .ITOUTO:NNBBL TROUBLE NANOBUBBLE INTRODUCTION: I. ntefilso ulainsineadbbl research, bubble and science nucleation of fields the In xeietleiec o aoube a rtpub- first was nanobubbles for evidence Experimental h rul ihnnbblsi that is nanobubbles with trouble The h rtpiti uhasrn ulttv contradic- qualitative strong a such is point first The .teciia aisi nusal qiiru point equilibrium unstable an is radius critical the 1. .temaue otc nl o yia nanobubbles typical for angle contact measured the 4. typical that predicts equation Laplace-Young the 3. param- realistic for radius critical conventional the 2. i sasdl on) hra aoube appear nanobubbles stable whereas be point), to saddle a is (it ue o arsoi rpeso h aesurface same mea- the on angle droplets contact macroscopic the for sured than larger significantly is atmospheric than larger or magnitude one pressure pressure of gas orders internal two an have nanobubbles the than curvature larger of so radius or nanobubble magnitude of typical order an is eters 3,4,9,12–17 ssal n httettletoyi aiu ttecritic the at maximum a is entropy hemisphe total immobilized the an that that and hemisph shown stable a is is nor it bubble stable, spherical are free non-class surface a two neither supersaturatio of Whilst increasing result radius. with combined decreases the tension be surface to shown is stability r,ta h rtclrdu sbt irsoi n thermo and microscopic both is radius critical the that ory, h bevdsaiiyo aoube otait h well the contradicts nanobubbles of stability observed The 1 n ic hntereitnehas existence their then since and o eetrve fter and theory of review recent a For hroyai tblt fNanobubbles of Stability Thermodynamic 19,20 is,teciia ra- critical the First, [email protected] 2–4 5–11 includ- Dtd .415) 3.14 (Dated: hlAttard Phil and yny Australia Sydney, , I.1 oa nrp fashrclarbbl nwtra a as surf a water and in 2 0 of bubble of ratio air supersaturation tension a spherical for a radius of of function entropy Total 1: FIG. qiiru path, equilibrium h iuieeulbimpath, equilibrium diffusive the otdcreflostecntn ubrpath, number constant the follows curve dotted yai eut r ati em fetoy hs can These entropy. of terms in thermo- cast all are article a this results to (In dynamic entropy. corresponds the point of equilibrium maximum local stable A bubble. ical to unlikely other. each are of points independent four be These and other data, wisdom. vari- simulation conventional in at computer measurements molecular-level is experimental systems, point diverse on broad with inapplicable which ance is ther- scales, surface, length macroscopic and nanoscopic conventional bulk that both again modynamics, prove point ex- to fourth the The appears than lifetimes. shorter nanobubble observed magnitude a whichperimentally of of orders dissolution atmosphere, time-scales runaway the on the the and to between lead nanobubble should gradient the that thermodynamic inside says strong point gas a third The is measured there nanobubbles. the of and scales thermodynamics values length by parameter predicted conventional bubbles using the for between scales discrepancy con- length quantitative question second a into The represents calls point itself. it thermodynamics that nucleation ventional nanobubbles of days) or radius. critical the marks line dotted The

iue1sostettletoyfrancetn spher- nucleating a for entropy total the shows 1 Figure Stot /kB - - -3.E+08 -2.E+08 1.E+081.E+08 0.E+00 1.E+08 2.E+08 ,adta hsgvsannsoi critical nanoscopic a gives this that and n, konrsl ncasclnceto the- nucleation classical in result -known yaial ntbe eenanoscopic Here unstable. dynamically rclbbl oieo nhydrophobic an on mobile bubble erical .E0 .E0 .E0 .E0 .E0 2.5E-06 2.0E-06 1.5E-06 1.0E-06 5.0E-07 0.0E+00 ia ubewt indcnatrim contact pinned a with bubble rical clmcaim.I ssonta the that shown is It mechanisms. ical lradius. al . 7 /.Tesldcreflostemechanical the follows curve solid The N/m. 072 ∂S tot /∂R ,tedse uv follows curve dashed the 0, = ∂S R (m) tot /∂N ,adtedash- the and 0, = N = N crit ace . 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. When the bubble is in mechanical equilibrium, the Laplace-Young equation says that the internal pressure 0.E+00

exceeds the external pressure. Since the pressure is a B k monotonic function of the chemical potential, and since / tot the external pressure is equal to the saturation vapor S pressure, the chemical potential of the air inside the bub- -1.E+08 ble must exceed the saturation chemical potential. This is the reason that a bubble in mechanical equilibrium can be in diffusive equilibrium only with a supersaturated -2.E+08 22 solution. 0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-06 2.5E-06 The critical radius is the point at which diffusive and R (m) mechanical equilibrium simultaneously hold. Using the the ideal gas equation of state, p = ρk T , and substitut- s B FIG. 2: Total entropy of a spherical air bubble in water ing the diffusive equilibrium equation into the mechanical as a function of radius following the mechanical equilib- equilibrium equation, the critical radius is rium path. The solid curves are for a supersaturation ra- tio of s = 2 and the dashed curves are for s = 3. The 2γ lower curves of each pair use the saturation surface tension Rcrit = . (2.8) (s 1)p γ† = 0.072 N/m, and the upper curves of each pair use the − supersaturated surface tension (spinodal supersaturation ra- The critical radius scales linearly with the surface tension tio s‡ = 5), γ(2) = 0.054 N/m, and γ(3) = 0.036 N/m. The and decreases with increasing supersaturation ratio. The arrows mark the respective critical radii (local entropy mini- depth of the entropy minimum (equivalently, height of mum). the free energy barrier) scales with the cube of the critical radius, S = k (s 1)ρ†V /2. tot,crit B crit the decrease is linear,23–25 The numerical− results− in Fig. 1 show that the criti- ‡ cal radius is a point of unstable equilibrium, which is † s s 19,20 γ(s) = γ − of course well-known in classical nucleation theory. s‡ 1 See Appendix B for a mathematical proof of the re- − p‡ p sult. Also this, the conventional formula for the criti- † g g = γ ‡ − † . (3.1) cal radius, would require a supersaturation ratio s =10– pg pg − 100 (equivalently, water in equilibrium with air at 10– Here γ† =0.072N/m is the saturation or coexistence sur- 100 atmospheres) in order to produce a critical radius † −7 −8 face tension (it was denoted plain γ above), s pg/pg Rcrit = 10 –10 m, which are the radii measured for ≡ nanobubbles. is the reservoir supersaturation ratio, which is the ratio of the supersaturation air vapor pressure to the satu- ration air pressure, and the dagger and double dagger denote the coexistence and the spinodal values, respec- III. STATE DEPENDENT SURFACE TENSION ‡ † ‡ † tively. Obviously, s >s = 1 and pg >pg. Simulations of a Lennard-Jones fluid at various temperatures found As explained above, if a bubble is in diffusive equilib- that the surface tension was well-fitted by this with spin- rium with the water phase, then the solution must be odal supersaturation ratios in the range s‡ = 3–6.23–25 supersaturated with air.22 The effects of supersaturation This surface tension may be called the supersaturated on the surface tension are neglected in the classical the- surface tension. It has a constant value independent of ory, but in fact these effects can be quite dramatic. the radius of the bubble and dependent only on the su- The barrier to bubble growth is the depth of the en- persaturation ratio of the reservoir. (the concentration tropy at the critical radius, which scales with the cube of air dissolved in the water divided by the saturation of the surface tension. At the spinodal supersaturation, concentration). there is no barrier to bubble growth, which means that If one uses this reservoir state-dependent surface ten- the surface tension must vanish.20 Hence the surface ten- sion, then all of the results of classical nucleation theory sion must decrease with increasing supersaturation.23–25 hold, with the value of the surface tension being replaced This general thermodynamic result is consistent with by the supersaturated value, γ† γ(s). This means that what is known experimentally from high pressure mea- both the critical radius, which⇒ scales linearly with the surements, namely that the surface tension of water is surface tension, and the nucleation barrier, which scales lower in the presence of oxygen or nitrogen and water cubicly with the surface tension, are reduced from their vapor than in the presence of pure water vapor.26 classical values. Computer simulations of the supersaturated liquid- Figure 2 shows the effect of the supersaturated sur- vapor interface show that to a reasonable approximation face tension on the total entropy of a bubble. It can be 4

1.E-04 tension with increasing supersaturation appears to solve one of the two major problems that nanobubbles pose for 1.E-05 conventional thermodynamics. As has been previously argued,23 it explains how the equilibrium radius comes 1.E-06 to be of nanoscopic dimensions. (m) crit R R 1.E -07 IV. HEMISPHERICAL BUBBLE ON A SURFACE 1.E-08

A. Mobile Hemispherical Bubble 1.E-09 1 2 3 4 5 s For a bubble on a solid surface, the optimum exterior contact angle is given by the Young equation

FIG. 3: Critical radius (log scale) as a function of supersatura- γsg γsl tion for an air bubble in water. The solid curve uses the satu- γsl = γsg + γlg cos(π θc), or cos θc = − . (4.1) − γlg ration surface tension, γ† = 0.072 N/m, and the dashed curve ‡ uses the supersaturation surface tension, γ(s), with s = 5. The liquid-vapor surface tension γlg was variously de- noted γ, or γ†, or γ(s) above. For a hydrophobic surface, which is the present concern, the surface energies of the seen that the critical radius and the nucleation barrier solid interfaces satisfy γsg < γsl, or θc > π/2. Define are indeed reduced from their conventional values. The ∆γ γsg γsl < 0. ≡ − critical radius is nanoscopic for realistic parameter val- It should be noted that the difference in the solid sur- ues, and would move to even smaller values for larger face energies is assumed fixed with a value determined reservoir supersaturation ratios, or if the spinodal super- from measurements on macroscopic water droplets on the saturation ratio had a smaller value. The critical radius given surface and the saturation surface tension. Con- is plotted as a function of supersaturation in Fig. 3. sequently the optimum contact angle for nanobubbles, Explicit comparison can be made with measurements which is given by the Young equation, increases with in- on nanobubbles. A supersaturation ratio of s =3.6 and creasing supersaturation ratio since the liquid-vapor sur- a spinodal supersaturation of s‡ = 5 gives a supersatu- face tension decreases with increasing supersaturation ra- ration surface tension of γ(s)=0.026 N/m and a critical tio. This notion is the origin of the values of the surface radius of Rcrit = 200 nm. Alternatively, s = 4.4 gives tension given above, which were deduced from the mea- 18 γ(s)=0.010 N/m and a critical radius of Rcrit = 60 nm. sured contact angle for nanobubbles. These are comparable to values obtained from measure- Consider an air bubble with radius of curvature R, ments on nanobubbles, specifically from analysis of the apex height above the surface of h, and contact rim radius difference between the nanoscopic and macroscopic con- r. The contact rim radius and apex height are related by 18 tact angles, with the former being obtained from from r2 + h2 2Rh =0. (4.2) the tapping mode image profiles of nanobubbles.3,4 In one − series of experiments the average radius of curvature was Hence r = √2Rh h2 or h = R √R2 r2. R = 200nm, and a surface tension of γ(s)=0.015 N/m The interior contact− angle is θ−′ = π −θ, and the an- was deduced from the contact angle. From another se- gle between the surface and the curvature− radius at the ries, R = 60 nm and γ(s)=0.019 N/m were obtained. contact rim is φ = (π/2) θ′ = θ π/2. One has An alternative interpretation of these results is discussed (R h)/R = sin φ = cos θ−, or cos θ =− 1+ h/R. in the conclusion to the paper. The− volume of the− bubble is − From the curves in Fig. 2 it can be seen that the critical 1 radius remains an entropy minimum when the supersat- V = π Rh2 h3 , (4.3) urated surface tension is used. Hence the nanobubble  − 3  remains unstable. In Appendix C the effect of using the and the area of the liquid-vapor interface is local supersaturation (ie. the supersaturation in the solu- 2 2 tion adjacent to the air-water interface, which is in diffu- Alg =2πR R R r =2πRh. (4.4) sive equilibrium with the air inside the bubble), which is h − p − i greater than the reservoir supersaturation, is discussed. The area of the solid-gas interface is of course Asg = Even in this case, and for a general non-linear model of πr2 = π(2Rh h2). − the supersaturated surface tension, it is proven that the The total entropy is nanobubble is always unstable. Hence in general the su- µ p S = S (N,V,T )+ N V persaturated surface tension alone cannot give a stable total s T − T nanobubble. γ ∆γ lg A A . (4.5) Despite the lack of stability, the decrease in surface − T lg − T sg 5

Again one can use the ideal gas entropy for the bubble, 3.E+07 S (N,V,T ) = Nk [1 ln NΛ3/V ]. The total entropy s B 2.E+07 is most conveniently written− as a function of N, R, and either h or r. 1.E+07

Using Stotal(N,R,h T ), the derivatives are B k | / 0.E+00 tot S S ∂Stotal µs µ = − + , (4.6) -1.E+07 ∂N T T -2.E+07 ∂S p p ∂V (R,h) total = s -3.E+07 ∂R h T − T i ∂R 0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-06 2.5E-06 γlg ∂Alg(R,h) ∆γ ∂Asg(R,h) R (m) − T ∂R − T ∂R ps p 2 γlg + ∆γ FIG. 4: Total entropy of a mobile hemispherical air bubble in = πh 2πh, (4.7) − T − T − T water on a hydrophobic surface, ∆γ = 0.036 N/m, following h i the mechanical equilibrium path. The solid curves are for a and supersaturation ratio of s = 2 and the dashed curves are for s = 2.5. The lower curve of each pair uses the coexistence sur- face tension γ† = 0.072 N/m, and the upper curve of each pair ∂Stotal ps p ∂V (R,h) = uses the supersaturated surface tension γ(2) = 0.054 N/m, ∂h h T − T i ∂h and γ(2.5) = 0.045 N/m. γlg ∂Alg(R,h) ∆γ ∂Asg(R,h) − T ∂h − T ∂h p p = s π 2Rh h2 TABLE I: Geometry of a Mobile Hemispherical Bubble at T − T − the Critical Radius. The difference in solid energies is ∆γ = h i   γ ∆γ −0.036 N/m. lg 2πR 2π[R h]. (4.8) − T − T − s γ Rcrit rcrit hcrit θc If the total entropy is a maximum simultaneously with N/m (µm) (µm) (µm) (deg.) respect to curvature radius R and apex height h, then the left hand sides of Eqs (4.7) and (4.8) are zero. These 2.0 0.072† 1.44 1.25 0.720 120 ∗ two equations have simultaneous solution 2.0 0.054 1.08 0.805 0.360 132 2.0 0.048# 0.960 0.635 0.240 139 R h ∆γ = γ − = γ cos θ, (4.9) 2.5 0.072† 0.960 0.831 0.480 120 − R 2.5 0.045∗ 0.600 0.360 0.120 143 # which is the contact angle condition, and 2.5 0.036 0.480 0. 0. 180 3.0 0.072† 0.720 0.624 0.360 120 2γ ∗ ps = p + , (4.10) 3.0 0.036 0.360 0. 0. 180 R 3.0 0.024# 0.240 0. 0. > 180 which is the Laplace-Young equation. These two results †Coexistence value. hold whether or not the entropy is maximal with respect ∗Supersaturated value using s‡ = 5. to number (ie. whether or not Eq. (4.6) is zero). #Supersaturated value using s‡ = 4. In Fig. 4 the total entropy is plotted along the path of mechanical equilibrium, which is equivalent to imposing the Laplace-Young equation and the contact angle con- dition on the bubble as it grows. It can be seen that B. Pinned Hemispherical Bubble the total entropy is a minimum at the critical radius. The geometric parameters of a bubble at the critical ra- For the case of a hemispherical bubble on a planar sub- dius for these and other thermodynamic parameters are strate, if the contact rim of the bubble is mobile, (ie. as it given in Table I. An optimum contact angle of 180◦ or is when it satisfies the contact angle condition) then its greater means that it is favorable for a planar gas layer volume increases with increasing radius of curvature, just to form between the solid surface and the liquid phase as for a free spherical bubble. In both cases the critical (drying transition). The results in Fig. 4 confirm that a radius is unstable. However if the contact rim is pinned hemispherical bubble mobile on a surface has the same then an increase in the radius of curvature decreases the stability characteristics as a free spherical bubble, which volume, and vice versa. In this case the critical radius is to say that it remains unstable. becomes a point of stable equilibrium. 6

The physical origin of the stability is readily under- 2.E+05 stood. Following closely the discussion of the instabil- ity of a free bubble given in the third last paragraph of 1.E+05 the introduction, at the critical radius of curvature for a pinned bubble, a fluctuation to a larger radius of curva- B k ture along the path of mechanical equilibrium reduces the / 0.E+00 tot S internal pressure and hence the gas chemical potential in- S side the bubble. There is now a thermodynamic gradient that forces dissolved gas from the reservoir into the bub- -1.E+05 ble, which increases the internal pressure. This creates a mechanical force imbalance, which acts to increase the -2.E+05 volume of the bubble at the expense of the volume of the 0.0E+00 5.0E-07 1.0E-06 1.5E-06 reservoir. So far this is the same as for a free bubble and R (m) for a mobile adhering bubble. But the difference for a pinned bubble is that an increase in volume reduces the FIG. 5: Total entropy of a pinned hemispherical bubble radius of curvature, which counteracts the original fluc- as a function of the radius of curvature for fixed contact tuation and restores the bubble to its original state. The rim radius r = 10−7 m. The solid curves follow the me- converse occurs if the fluctuation is to a smaller radius of chanical equilibrium path and dashed curves follow the dif- curvature. Hence the critical radius is a point of stable fusive equilibrium path. The lower pair of curves are for equilibrium for a pinned bubble. s = 2 and γ(s) = 0.054 N/m, the middle pair use s = 3 This physical argument that the pinned critical bubble and γ(s) = 0.036 N/m, and the upper pair use s = 4 and is at an entropy maximum can be readily confirmed by γ(s) = 0.018 N/m (s‡ = 5 in all cases). The arrows denote numerical calculation. First one needs to derive the en- the entropy maxima, which are at the critical radii. tropy derivatives, the vanishing of which give the critical radius. For a pinned (immobile) bubble attached to a hydrophobic surface, the rim radius r is constant. Hence and needs to use r instead of h as an independent variable so ∂Stotal(N, R r, T ) that r can be held fixed during the derivatives with re- | ∂R spect to R and N. Using the fact that h = R √R2 r2, p p ∂V (R, r) γ ∂A (R, r) the volume of the bubble may be written − − = s lg lg . (4.17) h T − T i ∂R − T ∂R 1 V = π Rh2 h3 (4.11) Hence at the bubble stationary point when the radius  − 3  derivative vanishes (mechanical equilibrium for a pinned 2 2 1 contact rim) one has = π R3 R2 R2 r2 r2 R2 r2 , 3 − 3 − − 3 −  p p ∂Alg(R, r) ∂R ps = p + γlg and the area of the liquid-vapor interface is ∂R ∂V (R, r) 4√R2 r2 4R +2r2/R 2 2 = pr + γlg − − . (4.18) Alg =2πR R R r . (4.12) 2R√R2 r2 2R2 + r2 h − p − i − − 2 If R r, then p p + 2γ /R, which is the usual The area of the solid-gas interface is of course Asg = πr . ≈ s ∼ r lg The derivatives are Laplace-Young expression. More generally, the left hand side is ps = NkBT/V (R, r). Inserting this into the above ∂V (R, r) 4πR π[2R2 + r2]R and taking the volume over to the right hand side gives s =2πR2 R2 r2 , 2 2 an explicit equation for N(R) along the path of mechani- ∂R − 3 p − − 3√R r − (4.13) cal equilibrium. This may be inserted into the expression for the total entropy and the latter plotted as a function 2 ∂Alg(R, r) 2 2 2πR of the radius of curvature for R > r. For the case of diffu- =4πR 2π R r , (4.14) † 2 2 sive equilibrium, N = sρ V (R, r). Either the saturated ∂R − p − − √R r g − surface tension γ† or the supersaturated surface tension γ(s) may be used. ∂A (R, r) sg =0. (4.15) Figure 5 shows the total entropy as a function of the ∂R radius of curvature when the contact rim is pinned at r = 10−7 m. Unlike the previous cases, the total entropy With these, the derivatives of the total entropy at fixed is now a maximum at the critical radius. The maximum r are is rather broad and extends down almost to the pinned ∂S (N, R r, T ) µ µ radius itself. For supersaturations of s = 3 and s = 4, total | = − s + , (4.16) ∂N T T the total entropy maximum is positive, which indicates 7

ters (actual supersaturation ratio of the solution and the -136620 spinodal supersaturation ratio). The origin of the super- saturation has been variously attributed to the heating -136640 of the solution and the surface by the laser diode, the -136660 piezo-electric crystal, and exothermic mixing during the B -136680 exchange of ethanol for water, and also to the entrain- /k 18 tot tot tot tot -136700 ment of air by the surfaces. In any case, whatever the S -136720 origin of the air, once stable nanobubbles are present, the -136740 inescapable thermodynamic conclusion is that the solu- tion is supersaturated. -136760 The new contributions here on this point were the ex- -136780 ρ plicit calculations for free and adsorbed bubbles, and the R mathematical proof that the state-dependent surface ten- sion on its own can never give a stable nanobubble, Ap- FIG. 6: Total entropy of a pinned hemispherical bubble (s = pendix C 2. 2, γ(s) = 0.054 N/m, r = 10−7 m) in the neighborhood of On the first point concerning stability, it was confirmed the critical radius of curvature and density (±20% in each that neither a free spherical bubble nor a mobile hemi- dimension). spherical bubble on a solid surface were stable. However, if the bubble was pinned with fixed contact rim, then it became stable. The physical reason for the change that it is favorable to have a pinned bubble compared to in stability is that whereas both a free spherical bubble no bubble at all. and a mobile hemispherical bubble have a volume that Figure 6 is a surface plot of the total entropy for the increases with increasing radius, the volume of a pinned pinned hemispherical bubble over the curvature radius- hemispherical bubble decreases with increasing radius of density plane. It can clearly be seen that the critical curvature. This is sufficient to confer stability. radius and density is a point of global entropy maximum. The fourth point is resolved by either or both of these In other words the pinned (immobile) bubble is stable. non-classical notions. The decrease in surface tension from the saturated value of a macroscopic droplet will increase the contact angle provided that the solid surface V. CONCLUSION energies remain unchanged. This observation was the ori- gin of the estimate of the surface tension of nanobubbles In the introduction four contradictions between classi- given in Ref. 18, and it appears to be consistent with their cal nucleation theory and nanobubbles were pointed out: measured radii of curvature. However, if the nanobubble (1) in the classical theory bubbles are unstable whereas is pinned, then the contact angle will also increase as the nanobubbles appear stable, (2) the classical critical ra- radius of curvature grows toward its critical value. dius is 1–2 orders of magnitude larger than the nanobub- For example, for the case of a macroscopic contact an- ◦ ble radius of curvature, (3) the classical internal pressure gle of 120 , (∆γ = 0.036 N/m), a mobile bubble using − † of nanobubbles is 1–2 orders of magnitude larger than the saturated surface tension γ = 0.072 N/m has the atmospheric pressure, and (4) the macroscopic contact same contact angle at the critical radius. At s = 2 angle is significantly larger than the measured nanobub- and γ(s) = 0.054 N/m, a mobile bubble has contact ◦ ble contact angle. angle 132 at the critical radius. A bubble pinned at ◦ † It was shown here that two non-classical ideas are r = 100nm has contact angle 176 for γ = 0.072 N/m, ◦ necessary and sufficient to account for the behavior of and 172 for γ(s)=0.054N/m, each at the respective nanobubbles, namely the dependence of surface tension critical radius (entropy maximum). on supersaturation, and the pinning of the contact line Pinning evidently has a more dramatic effect on the of the nanobubble on the surface. contact angle than supersaturation. The fact that the It has previously been pointed out that bubbles can measured contact angles for nanobubbles are larger than only be in diffusive equilibrium with a supersaturated that of macroscopic water droplets on the same surface solution,22 and that the supersaturated surface tension is supports both non-classical notions and on its own can- smaller than the saturated surface tension.23 These facts not decide between them. Conversely, the calculations in reduce the size of the critical radius and the magnitude Ref. 18 of the surface tension of nanobubbles from the of the internal pressure from those predicted by classical measured contact angles are probably not quantitatively nucleation theory. This largely resolves the second and reliable because they neglect the possibility of pinning. third problems enumerated above since, depending upon Two questions now arise: is pinning the only mecha- the parameters, it is possible to get quantitative agree- nism that will confer stability to a bubble? And what is ment with the measured radius of curvature of nanobub- the physical origin of pinning? bles and the deduced value of the surface tension18 (see It has previously been shown that a finite one- below) for apparently reasonable values of the parame- component system leads to stable bubbles,22 and it is 8

bile bubble would be expected to have a circular con- tact line. The nanobubbles are negatively charged and their morphology changes with pH. Although the electric double layer repulsion between neighboring nanobubbles would prevent their contact lines from growing, it would not prevent them from shrinking. One concludes that it is probably not a general a source of stability. Even in the absence of permanent heterogeneities or FIG. 7: Sketch of a nanobubble pinned by its deformation of neighbor interactions, it is possible that self-induced pin- the solid surface (not to scale). ning of the contact line may occur due to the elastic or viscoelastic deformation of the substrate by the nanobub- of interest whether similar results hold for nanobubbles ble, as is sketched in Fig. 7. The deformation is due to derived from air dissolved in water. A model of a finite the twin effects of the excess pressure inside the nanobub- reservoir based upon the limited diffusion of air in water ble and the tension of the air-water interface applied at in investigated in Appendix D. This model does yield an the contact line. Depending upon the relaxation time of entropy maximum and stable bubbles at radii beyond the the substrate, the barriers to relaxation, and the driving classical critical radius, but only on unrealistically short forces for horizontal motion, the contact rim may be effec- time scales. It is concluded that on experimental time tively pinned at the deformation. It is clear that the cost scales the reservoir of dissolved gas is effectively infinite of deformation of the base and rim of the substrate favor due to diffusion. Hence it appears that pinning is the a small contact radius, whereas the cost of curvature of only viable source of stability for nanobubbles. the contact rim favors a large contact radius. The latter The most obvious sources of pinning are permanent effect is much more significant for a nanobubble than for heterogeneities on the surface. However, most measure- a macroscopic bubble. ments on nanobubbles are performed on surfaces deliber- In conclusion, this paper has shown that two non- ately chosen to be smooth and chemically homogeneous. classical notions are necessary and sufficient to account This does not rule out random heterogeneities in any for the size and stability of nanobubbles: the surface ten- one case, particularly when isolated nanobubbles are ob- sion depends upon the level of supersaturation of the served. Because nanobubbles are so small, a very low solution, and pinning of the contact rim confers thermo- surface density of heterogeneities may be sufficient to pin dynamic stability. the contact line at a few points, which would immobilize These appear to solve the theoretical thermodynamic the whole rim. Conversely, pinning at a few nanoscopic problems posed by nanobubbles. They indicate the direc- heterogeneities likely has negligible effect on macroscopic tion for more experimental work to further characterize bubbles or droplets. nanobubbles and to establish that control over the phe- Another possible source of pinning is the electric dou- nomena that is a prerequisite for industrial or technolog- ble layer repulsion between neighboring nanobubbles. ical exploitation. Quantitative measurements with con- The images of nanobubbles obtained by Tyrrell and trolled supersaturation, either by means of de-aeration, Attard3,4 show them to be close packed on the surface, over-gassing, or temperature control, represent one line exactly as one would expect for the surface nucleation of enquiry. Measurements with substrates of different from a supersaturated solution. Further, the contact deformability, roughness, or polymer extensivity are also lines are often irregular and non-circular, whereas a mo- suggested by the present results.

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E. S. Kooij, and D. Lohse, Langmuir 23, 7072 (2007). Appendix B: Unstable Critical Radius 18 P. Attard, Eur. Phys. J. Special Topics 223, 893 (2014). 19 J. S. Rowlinson and B. Widom, Molecular Theory of Cap- As was shown explicitly in Fig. 1 above, the critical illarity, (Oxford University Press, Oxford, 1982). 20 radius is a saddle point, which is an unstable equilibrium D. W. Oxtoby, Acc. Chem. Res. 31, 91 (1998). 21 P. Attard, Thermodynamics and Statistical Mechanics: point. This may be proven mathematically by showing Equilibrium by Entropy Maximisation, (Academic Press, that a bubble does not satisfy the general stability re- London, 2002). quirements. 22 M. P. Moody and P. Attard, J. Chem. Phys. 117, 6705 (2002). 23 M. P. Moody and P. Attard, Phys. Rev. Lett. 91, 056104 1. General Stability Condition (2003). 24 M. P. Moody and P. Attard, J. Chem. Phys. 120, 1892 Since the probability of a fluctuation in number or vol- (2004). ume is proportional to the exponential of the entropy, 25 S. He and P. Attard, Phys. Chem. Chem. Phys. 7, 2928 (2005). two conditions necessary for stability are that the corre- 26 R. Massoudi and A. D. King Jr., J. Phys. Chem. 78, 2262 sponding second derivatives of the entropy are negative, (1974). SNN < 0, and SV V < 0, (B.1) at least at a stable stationary point. More generally, Appendix A: Two-Component Formulation the necessary and sufficient condition for stability is that both eigenvalues of the Hessian matrix of second deriva- At the more fundamental level, a bubble and sur- tives are negative. The eigenvalues are determined from rounding liquid should be treated as two coexisting two- the characteristic equation, component mixtures, one rich in air, and the other rich S λ S in water. Let air be denoted by the subscript 1 and water NN − NV SVN SV V λ by the subscript 2. The total entropy is − 2 = [SNN λ][SV V λ] SVN Stot(N1,N2, V µ1,µ2,p,T ) 2 − − − 2 | = λ λ [SNN + SV V ]+ SNN SV V SNV ,(B.2) µ1 µ2 pV γA − − = Ss(N1,N2,T )+ + . (A.1)   T T − T − T whose roots are Equilibrium is characterized by the vanishing of the 1 λ± = [S + S ] (B.3) derivatives, namely 2  NN V V

µs1 = µ1, µs2 = µ2 (A.2) 2 2 [SNN + SV V ] 4 [SNN SV V S ] . ± q − − NV  and 2γ The argument of the square root is always non-negative. p = p + . (A.3) The only way for both eigenvalues to be negative is if s R the square root is smaller than the magnitude of the first At saturation one has term, which occurs when the second bracketed term in † † the square root is positive. Hence in addition to the p = p + p , (A.4) 1 2 negativity of the two pure second derivatives, stability where p is atmospheric pressure. For a supersaturation is ensured by the positivity of the determinant of the ratio of s, the liquid solution phase (ie. the reservoir) is Hessian matrix, † in equilibrium with a gas phase of partial air pressure sp1 S S S2 > 0. (B.4) † NN V V − NV and partial water vapor pressure of p2. When the bub- ble is in diffusive equilibrium with the solution, the sum of these must give the internal pressure of the bubble. 2. Bubble Instability Hence the critical radius is given by 2γ Specifically for a bubble, the second derivatives of the Rcrit = † † total entropy are sp + p p 1 2 − 2γ kB = . (A.5) SNN = − , (B.5) (s 1)p† N − 1 The difference between this and the one-component for- ρkB 2γ 1 mulation is that the partial vapor pressure of air at sat- SV V = − + 2 2 † V T R 4πR uration, p1 0.8p, has been replaced by atmospheric p 2γ ≈ = − s + , (B.6) pressure, p. VT 3V T R 10 and The derivative with respect to radius is

kB ∂Stotal ps pr ∂V (R) SNV = . (B.7) = V ∂R h T − T i ∂R γ ∂A(R) γ′(ρ)A ∂ρ(R,N) Clearly SNN is always negative. At the equilibrium − T ∂R − T ∂R point, the Laplace-Young equation shows that SV V = ps pr 2 (pr/VT ) (2γ/V T R)+(2γ/3VT R), which is negative. = 4πR These− two− conditions are necessary but not sufficient con- h T − T i γ(ρ) 12πRρ ditions for the equilibrium point to be stable. 8πR + γ′(ρ) . (C.3) The determinant of the Hessian matrix is − T T k2 1 2γ This vanishes when S S S2 = B − ρ + 1 NN V V − NV V 2  ρ − 3k T R −  2γ(ρ) 3ργ′(ρ) B p = p + . (C.4) 2 s r kB 2γ R − R = − 2 , (B.8) V 3ρkBT R Like the usual Laplace-Young equation, this says that the internal pressure of the air bubble is larger than the exter- which is always negative. As shown above, it is necessary nal pressure of the liquid, since γ(ρ) 0, and γ′(ρ) < 0. for stability that this determinant be positive. Since the usual Laplace-Young equation≥ uses the coexis- One concludes from this that the bubble’s stationary tence value for the surface tension, γ† γ(ρ†) γ(ρ), point, the critical radius Rcrit and density ρcrit, is a saddle whether the internal pressure predicted≡ by the≤ present point, which is a point of unstable equilibrium. theory is larger or smaller than that predicted by the constant surface tension theory depends on the specific surface tension function. Appendix C: Local Supersaturation

1. Critical Radius for Local Supersaturation 2. Instability of the Critical Radius for Local Supersaturation The simulations that justify the formula for the super- saturated surface tension were carried out with diffusive At the bubble critical radius (ρcrit, Rcrit), both deriva- equilibrium established across the interface. In homoge- tives vanish, SN (ρcrit, Rcrit) = 0 and SV (ρcrit, Rcrit) = 0. neous nucleation theory it is more traditional to plot the Individually they give a non-linear function for the opti- total entropy (equivalently free energy) as a function of mum density radius along the path of mechanical equilibrium. In this ′ † − B case diffusive equilibrium does not hold except at the crit- ρ = sρ e 3γ (ρ)/k TR, (C.5) ical radius. One could argue that there is always a local diffusive equilibrium in the immediate locality of the bub- and ble air-water interface, and that this is what determines ′ † 2γ(ρ) 3ργ (ρ) the value of the surface tension. In this case one would ρ = ρ + − . (C.6) k T R use γ(s ) γ(ρ), where s (ρ) ρ/ρ† p /p†, for B local ≡ local ≡ g ≡ s g the bubble on the path of mechanical equilibrium. The simultaneous solution of these gives the critical ra- If one uses this bubble-dependent model for the super- dius and density. If the derivative of the surface tension saturated surface tension, the total entropy is is negative, γ′(ρ) < 0, then the first of these says that the gas density inside the bubble is greater than the super- µrN prV saturation vapor density. The second says that the gas Stotal(N,R,T ) = Ss(N,V,T )+ T − T density inside the bubble is greater than the coexistence γ(ρ)A density, both terms of the correction being positive. , (C.1) − T Solving instead for the critical radius one obtains

† ′ with ρ = N/V and pr = ρgkBT . For full generality, and 3γ (ρ) ρ not R = − ln † , (C.7) to make an important point, it will here be assumed kBT sρ that γ(ρ) is a linear function of density. The number derivative is and 2γ(ρ) 3ργ′(ρ) ∂Stotal µs µr ′ A R = − . (C.8) = − + γ (ρ) . (C.2) k T [ρ ρ†] ∂N T T − VT B − † ‡ When this derivative vanishes, the chemical potential in- The second of these gives R(ρ) for ρ [ρg,ρg] along the side the bubble is higher than that outside. path of mechanical equilibrium. ∈ 11

Since the critical radius must vanish at the spinodal, The determinant should be evaluated at the bubble ‡ R(ρg) = 0, this says that critical point, which is denoted by an over-line. For typo- graphical simplicity, the density argument of the surface 3 tension will be suppressed. One has γ(ρ‡ )= ρ‡ γ′(ρ‡ ). (C.9) g 2 g g V 2 1 3γ′′ S S = It is not possible to satisfy this in the linear model, since 2 NN V V − kB  ρ − RkBT  the left hand side is positive or zero and the right hand 2γ 6ργ′ 9ρ2γ′′ side is strictly negative. Hence the local supersaturation- ρ + − − dependent surface tension must be a non-linear function × − 3RkBT  ′ † ′ 2 ′′ ′′ of the internal gas density. Since in general γ (ρg) < 0, 2γ 6ργ 9ρ γ 3ργ ′ ‡ = 1 − − + and this result says that γ (ρ ) 0, one must have that − 3RρkBT RkBT g ≥ 2γγ′′ 6ργ′γ′′ 9ρ2γ′′2 ′′ γ (ρ) 0, (C.10) − −2 (C.18) ≥ − (RkBT ) † ‡ assuming that the interval [ρg,ρg] is relatively small. and This result holds only for the local supersaturation- 2 ′ ′′ 2 dependent surface tension. It will now be shown that this V 2 γ 3ργ 2 SNV = 1+ + (C.19) result precludes the critical radius from being a point of kB  RkBT RkBT  stable equilibrium. 2γ′ +6ργ′′ 6ργ′γ′′ + γ′2 +9ρ2γ′′2 The second derivatives of the total entropy are = 1+ + 2 . RkBT (RkBT ) ′′ kB 3γ (ρ) For a stable bubble critical point, the determinant should SNN = − , (C.11) N − V T R be positive

2γ +6ργ′ 2γγ′′ 6ργ′γ′′ 9ρ2γ′′2 ′ 0 < − − − ρkB 2γ(ρ) 3ργ (ρ) 1 2 3RρkBT − (RkBT ) SV V = − + − 2 2 V T R 4πR ′ ′′ ′ ′′ ′2 2 ′′2 γ′(ρ)+3ργ′′(ρ) ρ 2γ +6ργ 6ργ γ + γ +9ρ γ 2 − T R V − RkBT − (RkBT ) ρk 2γ(ρ) 6ργ′(ρ) 9ρ2γ′′(ρ) 2γ 18ρ2γ′′ 2γγ′′ + γ′2 B = − − . (C.20) = − + − − (C.12), 2 V 3V T R 3RρkBT − (RkBT ) and This gives k 3γ′(ρ) 1 3γ′′(ρ) ρ 2 ′′ ′′ B ′2 2 2γ 18ρ γ 2γγ SNV = + + . (C.13) γ < (Rk T ) − − . (C.21) 2 2 B 2 V T R 4πR T R V  3RρkBT − (RkBT ) 

Clearly SNN is negative if This has a solution if, and only if, the bracketed term is positive, which means that ′′ kBT R γ (ρ) > − . (C.14) 2γ 2γγ′′ 18ρ2γ′′ 3ρ − > + 2 3RρkBT (RkBT ) 3RρkBT could ′′ Although this bound be satisfied by γ (ρ) = 0, this 2γ +6ρRk T possibility is ruled out by another bound below. = B γ′′, (C.22) 2 (RkBT ) Also SV V is negative if or 2γ(ρ) 6ργ′(ρ) 9ρ2γ′′(ρ) 0 > ρkBT + − − , (C.15) 2γ (Rk T )2 − 3R γ′′ < − B 3ρRkBT 2γ +6ρRkBT which can be written as Rk T/3ρ = − B . (C.23) ′ ′′ 3RρkBT +2γ(ρ) 6ργ (ρ) 1+3ρRkBT/γ γ (ρ) > − 2 − , (C.16) 9ρ Combining this with the condition found above for S < 0, γ′′ > Rk T/3ρ, gives or as NN − B 2 ′′ 3Rρk T +2γ(ρ) 9ρ γ (ρ) RkBT/3ρ ′′ RkBT γ′(ρ) > − B − . (C.17) < γ < . (C.24) 6ρ 1+3ρRkBT/γ − 3ρ 12

This rules out the possibility that γ′′ = 0, which is to 3.E+08 say that the local supersaturated surface tension cannot 2.E+08 be a linear function of density if a bubble is to be stable. 1.E+08 This agrees with the conclusion already made above on different grounds. 0.E+00 B k This result shows that the second derivative of the sur- / -1.E+08 tot S S face tension has to be negative if the free bubble is to -2.E+08 be stable. But Eq. (C.10) showed that the second den- sity derivative of the surface tension had to be positive -3.E+08 ′′ γ (ρ) > 0. These two results prove that it is not possi- -4.E+08 ble to confer thermodynamic stability on a bubble at the -5.E+08 critical radius by invoking a local supersaturation surface 0.E+00 1.E-06 2.E-06 3.E-06 4.E-06 5.E-06 tension. R (m)

Appendix D: Finite Reservoir FIG. 8: Total entropy of a spherical bubble in a finite reser- voir on the path of mechanical equilibrium (s = 2, γ† = 0.072 N/m). The solid curve is for an infinite reservoir, the In order to model the finite diffusion of air in water, one dashed curve is for a diffusion time t = 0.035 s, and the dotted can consider two reservoirs: an infinite volume reservoir curve is for t = 0.01 s. of atmospheric pressure p, and a reservoir of fixed volume Vr. The total volume Vtot = V + Vr is variable and can exchange with the infinite volume pressure reservoir. The where ρsol = Nr/Vr is the concentration of dissolved gas. Henry’s law may be written for the concentration of dis- total number of of air molecules is fixed, Ntot = N +Nr = spV /K, where K =2.3 10−19 J is Henry’s constant for solved gas in the form r × air in water. The idea is that due to limited diffusion over −1 −1 −3 µ/kBT ρsol = K p = K kBT Λ e , (D.6) a specified time scale, only air dissolved in the water of the fixed volume reservoir can exchange with the bubble. or One can define the fixed reservoir volume in terms of 3 KρsolΛ a diffusion length µ = kBT ln . (D.7) kBT 3 4πlD Vr = , lD √Dt, (D.1) Recall that the saturation air vapor pressure has been 3 ≡ taken to equal the external atmospheric pressure. Equat- where D =2 10−9 m2/s is the diffusion constant for air ing these two expressions for the chemical potential yields × and water, and t is some experimental time scale. K ε = k T ln . (D.8) The spherical bubble or sub-system is characterized B k T by number N, volume V =4πR3/3, and area A =4πR2. B With these the total entropy is Because the reservoir volume Vr is fixed and indepen- dent of the bubble, the derivative of the total entropy γ S = S (N,V,T )+ S (N , V ,T ) A with respect to radius vanishes when the Laplace-Young total s r r r − T p equation is satisfied, V + const. (D.2) − T tot 2γ ps = pext + . (D.9) The constant is chosen to make the entropy vanish at R zero radius. This gives the path of mechanical equilibrium. As usual the sub-system (bubble) is an ideal gas Figure 8 plots the entropy using this model for a fi- nite, diffusion-linked reservoir. The results are typical of 3 NΛ the model. For small bubbles the reservoir is effectively Ss(N,V,T )= kBN 1 ln . (D.3)  − V  infinite and all the curves coincide. For larger bubbles and small reservoirs (ie. short experimental time scales), The fixed volume reservoir containing the dissolved air the total entropy monotonically decreases and there is can be treated as an ideal gas in a uniform external field no critical radius. For larger reservoirs (ie. longer experi- 3 mental time scales), there is a critical radius and beyond NrΛ Nrε S (N , V ,T )= k N 1 ln . (D.4) this a stable radius where the entropy is a maximum. r r r B r  V  T − r − The decrease in entropy beyond this maximum is due to Here ε is the molecular solvation energy. The effective the depletion of the reservoir. Obviously, the radius of chemical potential that corresponds to this is maximum entropy will increase with increasing reservoir size. It should be noted that the typical lifetime observed 3 3 4 µr = ε + kBT ln ρsolΛ , (D.5) for nanobubbles is on the order of 10 –10 s.