Direct Measurement of the Surface Tension of Nanobubbles
aa r X i v : . [ c ond - m a t . s o f t ] M a y Direct Measurement of the Surface Tension of Nanobubbles
Phil Attard [email protected] , Sydney, Australia (Dated: 9 May, 2015)It is shown that when the nanobubble contact line is pinned to a penetrating tip the interfacebehaves like a Hookean spring with spring constant proportional to the nanobubble surface tension.Atomic force microscope (AFM) data for several nanobubbles and solutions are analysed and yieldsurface tensions in the range 0.04–0.05 N/m (compared to 0.072 N/m for saturated water), andsupersaturation ratios in the range 2–5. These are the first direct measurements of the surface tensionof a supersaturated air-water interface. The results are consistent with recent theories of nanobubblesize and stability, and with computer simulations of the surface tension of a supersaturated solution.
I. INTRODUCTION
Experimental evidence for nanobubbles was first pub-lished in 1994, and since then their existence hasbeen confirmed from various features of the mea-sured forces between hydrophobic surfaces, includ-ing a reduced attraction in de-aerated water, andfrom images obtained with tapping mode atomic forcemicroscopy. For a recent review of theory andexperiment, see Ref. 18.The initial controversies over the existence of nanobub-bles —that they should have an internal gas pressure of10–100 atmospheres that would cause them to dissolve inmicroseconds— have largely been resolved. First it wasshown that nanobubbles can only be in equilibrium inwater supersaturated with air, and then it was shownthat the surface tension of a supersaturated solution mustbe less than that of a saturated solution. Finally, itwas shown that nanobubbles with a pinned contact rimare thermodynamically stable. This means that con-tact line pinning is a necessary and sufficient conditionfor nanobubbles to be in mechanical and diffusive equi-librium with the supersaturated solution.The significance of the reduction in surface tension isthat the internal gas pressure, which can be calculatedfrom the Laplace-Young equation, is much less than thoseinitial estimates used to argue against nanobubbles. Italso means that the degree of supersaturation of the solu-tion, necessary for diffusive equilibrium of the nanobub-ble, is reduced to realistic levels that are attainable inthe fluid cell.For many it is surprising that purely on thermody-namic grounds (ie. no additives or surfactant) the surfacetension for nanobubbles should be reduced from the usualvalue of the air-water interface. Nevertheless, this resultis firmly established by thermodynamics, density func-tional theory, and computer simulation. Theresult is not widely acknowledged within the nanobub-ble field, possibly because many practitioners place moretrust in experimental measurement than they do in ther-modynamics or in mathematical equations. To close thisgap, it would be desirable to measure directly the surfacetension of nanobubbles.Measuring the surface tension of the supersaturated r r r FIG. 1: Hemispherical bubble penetrated and deformed by aconical tip (blunt radius r , height z ) upon which it sticksat r . air-water interface is a worthwhile experimental goal thathas application beyond nanobubbles. For example, in at-mospheric physics the nucleation of cloud droplets alwaysoccurs from a supersaturated atmosphere, and so the rateof change of surface tension with the degree of supersatu-ration is a key input necessary for quantitative modelingin that field.A supersaturated solution in the current context meansthat the concentration of air in the water is higher thanit would be if the water were equilibrated with air at thecurrent temperature and pressure. Supersaturation canbe readily achieved by previously equilibrating the waterwith air at a higher pressure or at a lower temperature.The difficulty in measuring directly the surface tensionof a supersaturated solution is that the change in surfacetension is determined by the concentration of air withinthe first nanometer of the interface, but, due to diffu-sion across the interface, this region is always saturatedrather than supersaturated, at least if the measurementis performed on a macroscopic droplet or bubble exposedto the atmosphere. In the case of nanobubbles, how-ever, the solution is supersaturated immediately in thevicinity of the interface due to the high internal gas pres-sure of the nanobubble, and so the surface tension of ananobubble must be that of a supersaturated solution.If one could measure the surface tension of the nanobub-ble, then one would have a means to determine the rateof change of surface tension with supersaturation. Thepresent author knows of no previous measurements of thesurface tension of the supersaturated air-water interface.The present paper is concerned with modelingatomic force microscope (AFM) force measurements onnanobubbles when the tip of the cantilever penetrates thenanobubble. The aim is to relate the surface tension ofthe nanobubble to the measured force. The results areapplied to force data from several nanobubbles. In allcases analyzed, the surface tension was always less thanthat of the saturated air-water interface, and the super-saturation ratio of the solution was always greater thanunity. II. PENETRATED HEMISPHERICAL BUBBLE
Figure 1 is a sketch of a hemispherical bubble on a solidsurface that is deformed by a penetrating conical tip.In a cylindrical coordinate system, the bubble contactsthe surface at ( r ,
0) and the tip at ( r , z ). The bubbleis pinned or fixed at these two radii. The blunt tip ofthe cone is at a height (separation) above the substrateof z for r ≤ r . The half angle is given by tan α =( r − r ) / ( z − z ). The case of a positive load, F > The ear-lier analysis explicitly included the effects of a surfaceor interaction force between the probe and the bubble,whereas here the bubble is penetrated by the probe andmakes either stick or slip contact with its sides. The ear-lier analysis was for a bubble mobile on the substrate andfor fixed number of air molecules, whereas here the bub-ble is pinned at r and is in diffusive equilibrium withthe supersaturated solution. Due to the present pinnedcontact rim, the solid surface energies and the contactangle play no role in the present analysis, whereas theydid for the case of a mobile bubble analyzed in Ref. 28.These differences in the model make the present analysisconsiderably simpler and shorter than that of the earliercase. Despite these differences in the model, the qualita-tive conclusion is the same in both cases, namely that thebubble interface behaves as a Hookean spring. The quan-titative expression for the spring constant found here is ofcourse specific for the present model of a bubble pinnedat r . A. Undeformed Bubble
As recently shown, once the bubble is pinned at thecontact radius r , it is thermodynamically stable at thecritical radius and density. Accordingly, the undeformedbubble is the critical bubble. It is of course hemispheri-cal, and its radius is the critical radius R c = 2 γ ( s − p = 2 γ ∆ p . (2.1) Here s > γ is the liquid-vaporsurface tension (of the supersaturated interface), p is the pressure of the reservoir (taken to be atmospheric),∆ p is the excess pressure of the bubble, k B is Boltzmann’sconstant, and T is the temperature. For simplicity, thereservoir pressure and the saturation vapor pressure areapproximated as equal. The undeformed bubble has volume V c = π (cid:2) R c z − z / (cid:3) , (2.2)liquid-vapor surface area A c = 2 πR c (cid:20) R c − q R − r (cid:21) = 2 πR c z c , (2.3)and apex height z c = R c − q R − r . (2.4)The undeformed profile is z c ( r ) = z c − R c + p R − r . (2.5)The undeformed bubble contains N c = spV c /k B T gasmolecules. In the present case of fixed contact rim r ,thermodynamic stability holds simultaneously for num-ber and volume fluctuations. Hence one does not haveto insist upon constant number as the author has had todo in previous work.
B. Deformed Bubble: Prick Stick
The bubble profile is z ( r ), which ends at the contactpoints ( r , z ) and ( r , r − r = [ z − z ] tan α , where α is thecone half angle, and z is the height of the tip of the tipabove the substrate, which is also called the separation.The tip is taken to be perfectly blunt, which is to say thatits end is a disc of radius r at z . Although in principleone could also have contact at ( r , z ) for r ≤ r , thiscase will be excluded in the numerical results below. Aperfectly sharp tip has r = 0.The volume is V [ z ] = 2 π Z r r d r rz ( r ) + πr z − V t = 2 π Z r r d r rz ( r ) + 2 πr α + πr h z − r tan α i − πr α . (2.6)The volume of the blunt tip inside the bubble is V t = πr α − πr α . (2.7)The terms πr z − V t give the vapor volume beneaththe tip, r < r . These may be neglected for the profiledifferentiation since they are constant. The fluid-vaporinterfacial area is A [ z ] = 2 π Z r r d r r p z ′ ( r ) . (2.8)The total entropy of the bubble and reservoir is a func-tional of the profile and a function of the other thermo-dynamic parameters, S tot [ z ] = S b ( N, V, T ) − γT A [ z ] − pT V [ z ] + µT N. (2.9)The solid-air and solid-liquid surface energies (for boththe substrate and the tip) are constant because of thefixed contact radii and so are neglected here. Here µ = k B T ln[ sp Λ /k B T ] is the chemical potential of theair. This expression is appropriate for the case that thebubble can exchange number, volume, and area with thereservoir.It is assumed that thermal equilibrium holds. The en-tropy of the bubble may be taken to be that of an idealgas for thermal equilibrium, S b ( N, V, T ) = k B N (cid:20) − ln N Λ V (cid:21) , (2.10)where Λ is the thermal wave length. One could infact use the entropy of a real gas instead of this sincethe only things that enter below are its thermody-namic derivatives, p b = T ∂S b ( N, V, T ) /∂V , and µ b = − T ∂S b ( N, V, T ) /∂N .Obviously, setting the number derivative of the totalentropy to zero yields the equilibrium condition µ b = µ ,which for an ideal gas is the same as N = spk B T V [ z ] . (2.11)Invoking the usual variational properties of equilibriumthermodynamics, the number is fixed at this value inall that follows.The functional derivative of the total entropy with re-spect to the bubble profile is δS tot [ z ] δz ( r ) = ∆ p T δV [ z ] δz ( r ) − γT δA [ z ] δz ( r ) . (2.12)where ∆ p ≡ p b − p = ( s − p is the excess pressure ofthe bubble. One has δV [ z ] = 2 π Z r r d r rδz ( r ) , (2.13)from which it follows that δV [ z ] δz ( r ) = 2 πr. (2.14) Also δA [ z ] = 2 π Z r r d r r z ′ ( r ) p z ′ ( r ) δz ′ ( r ) (2.15)= − π Z r r d r dd r " r z ′ ( r ) p z ′ ( r ) δz ( r ) , following an integration by parts and the vanishing of theperturbation at the boundaries. From this one has δA [ z ] δz ( r ) = − π dd r " rz ′ ( r ) p z ′ ( r ) . (2.16)Inserting these into the functional derivative of the totalentropy and setting the latter to zero gives a differentialequation (the Eular-Lagrange equation) for the optimumprofile, 0 = 2 πr ∆ p + dd r " πrz ′ ( r ) p z ′ ( r ) γ. (2.17)Due to diffusive equilibrium, the excess pressure is a con-stant, ∆ p = ( s − p .The first integral of this isconst. = πr ∆ p + 2 πrz ′ ( r ) p z ′ ( r ) γ. (2.18)
1. Force
Including a spring attached to the cantilever withspring constant k t , the extended total entropy is S tot ,k = S tot ([ z ] , r ) − k t T [ z − z c ] . (2.19)The cantilever spring is placed so that it is unextended(zero force) when the tip is just in contact with the un-deformed bubble.The derivative of the extended total entropy with re-spect to the tip position, at constant tip contact radius r , and evaluated at the optimum profile z ( r ; r , z ) isnow required.Above, in deriving the Eular-Lagrange equation forthe profile, the variation at the boundaries vanished, δz ( r ) = δz ( r ) = 0. In the present case, the variationat contact on the tip must be δz ( r ) = ∆ z . This meansthat one picks up an extra term from the integration byparts of the variation in area, δA [ z ] = 2 π Z r r d r r z ′ ( r ) p z ′ ( r ) δz ′ ( r ) (2.20)= − πr z ′ ( r ) p z ′ ( r ) ∆ z − π Z r r d r dd r " r z ′ ( r ) p z ′ ( r ) δz ( r ) . Accordingly
T ∂S tot ,k ∂z (cid:12)(cid:12)(cid:12)(cid:12) z ( r ) = (cid:20) ∆ p δV [ z ] δz ( r ) − γ δA [ z ] δz ( r ) (cid:21) z ( r ) ∂z ( r ) ∂z + πr ∆ p + 2 πr z ′ ( r ) p z ′ ( r ) γ − k t [ z − z c ]= 0 + F + F t . (2.21)The first term vanishes for the optimum profile. The re-maining term proportional to ∆ p arises from the deriva-tive of the constant contributions to the volume. Thisand the remaining term proportional to γ give the forceexerted by the bubble on the tip, F = πr ∆ p + 2 πr z ′ ( r ) p z ′ ( r ) γ. (2.22)This is just the pressure difference times the cross-sectioncontact area plus the vertical component of the surfacetension force times the contact perimeter.The quantity F t ≡ − k t [ z − z c ] is the force exerted bythe tip on the bubble. One sees that in the equilibriumor static case, when the extended total entropy is a max-imum, the force due to the bubble is equal and oppositeto the force due to the tip, F = − F t ( z ). A positive bub-ble force, as sketched in Fig. 1, corresponds to a negativecantilever force (in a signed sense). In practice, one oftencalls the cantilever force the applied force, or the load. Apositive cantilever force (negative bubble force) gives anextended rather than a flattened bubble.Comparing bubble force with the first integral,Eq. (2.18), one sees that the integration constant is justthe force exerted by the bubble on the tip, so that onehas F = πr ∆ p + 2 πrz ′ ( r ) p z ′ ( r ) γ. (2.23)Hence one has a differential equation for the profile as afunction of the force, F , the substrate contact radius r ,the tip contact radius r , and the tip contact height z .Given the geometry of the tip (eg. the half angle α ), thelocation of the tip of the tip, z , is determined by thelatter two quantities, if it is ever required.In what follows an analytic expression for the profilewill be derived in the weak force limit, The expansion isvalid when | F | R c / πγ ( s ) r ≪
2. Location of Dimple Rim
The dimple rim r is the maximum height of the bub-ble, so that z ′ ( r ) = 0. From the differential equation forthe profile this yields r = s Fπ ∆ p = s F R c πγ . (2.24) Note that since r ≥ r , this sets a lower limit on therepulsive force that gives rise to a dimple. Of course thereis also no dimple for attractive forces (negative loads). Ineither case, the dimple rim plays no further role in theanalysis.
3. Profile
Rearranging equation (2.23) for the profile, gives[ F − π ∆ p r ] [1 + z ′ ( r ) ] = (2 πγr ) z ′ ( r ) , (2.25)or z ′ ( r ) = ± [ F − π ∆ p r ] p (2 πγr ) − [ F − π ∆ p r ] . (2.26)The positive root is the physical root.One can see that a minimum and a maximum force isdefined for a given pinned tip contact line r when thegradient of the profile becomes infinite, z ′ ( r ) = ±∞ .The bubble will rupture when the applied load exceedsthese limits. From the profile equation one has F min = − πr γ (cid:20) − r R c (cid:21) , (2.27)and F max = 2 πr γ (cid:20) r R c (cid:21) . (2.28)In fact, since the gradient of the profile can’t be infiniteanywhere, these two limits hold for any r on the interval[ r , r ]. Using r gives the tightest upper bound because r ≥ r . But it can be the case that a tighter lower boundcan occur by taking r inside the interval. In particular,if r ≤ ( R c / ≤ r , then the bubble will rupture if F < − πR c γ/
2. If r ≤ R c /
2, then the bubble will rupture if
F < − πr γ [1 − r /R c ].For small loads, | F | ≪ πγ ( s ) r /R c , one can expandthe profile equation to linear order in the force, z ′ ( r ) = F − π ∆ p r p (2 πγr ) − ( π ∆ p r ) + 2( π ∆ p r ) F + O ( F )= ( F/π ∆ p ) − r r p R − r + 2( F/π ∆ p )= − r p R − r + F/π ∆ p r p R − r + rF/π ∆ p [ R − r ] / + O ( F ) . (2.29)Recall that ∆ p = 2 γ/R c . The first term gives the un-deformed profile, Eq. (2.5), and the remainder give theperturbation due to the force to linear order. With ε ( r ) ≡ z ( r ) − z c ( r ), this gives the derivative of the per-turbation to linear order, ε ′ ( r ) = F R c / πγr p R − r + rF R c / πγ [ R − r ] / . (2.30) H e i g h t , z ( r ) ( n m ) H e i g h t , Radial Coordinate, r (nm) FIG. 2: Pinned bubble profiles for various applied loads. Thedotted curve is the undeformed bubble (zero load), a full curveis an exact profile, and an adjacent dashed curve is the cor-responding linear approximation. Above and below the un-deformed profile the loads are ± . ± s = 4, γ ( s ) = 0 .
018 N/m ( s ‡ = 5), and r = 100 nm, giving R c = 120 nm, and z c = 54 nm. Eachcurve terminates at its last stable contact radius. The integral of this is ε ( r ) = F πγ (
12 ln 1 − √ − x √ − x + 1 √ − x − C ) , (2.31)where x ≡ r/R c . The integration constant is determinedby the condition that ε ( r ) = 0, C ≡
12 ln 1 − p − x p − x + 1 p − x . (2.32)Figure 2 shows several profiles of deformed bubbles.The exact profile was obtained by numerical integra-tion of the profile equation, Eq. (2.26). Here and belowthe linear approximation refers to analytic results basedthe expansion to linear order in the force, in this caseEq. (2.31). In the case of the figure, 2 πr γ/R c = 0 . r = 20 nm. For loads with magnitude much less thanthis the linear approximation can be guaranteed accurateaccurate. It can be seen in the figure that the perfor-mance of the linear approximation is rather better thanis indicated by this parameter. For larger loads there is asignificant discrepancy between the exact and the linearprofile. The problem is more acute for extensive than forcompressive forces.
4. Bubble Spring Constant
The various quantities z above were measured relativeto the substrate. Now, in the laboratory frame of refer-ence, let ζ s be the position of the solid substrate. In this ( e xa c t) / k b ( li n e a r ) k b ( e xa c t) Force, F (nN) FIG. 3: Ratio of exact effective bubble spring constant,∆ F/ ∆ z ( r ), to linear bubble spring constant k b , Eq. (2.37),as a function of the applied load. The solid, dashed, and dot-ted curves are for r = 10, 20, and 30 nm, respectively. Otherparameters as in Fig. 2. Each curve terminates at its limitsof solution. The inset magnifies the region around zero force. laboratory frame, the position of the tip, the tip contactcircle, and the undeformed bubble interface are ζ = z + ζ s , ζ = z + ζ s , and ζ c = z c + ζ s , (2.33)respectively. Initially, the substrate is at ζ s = 0, thetip is just touching the undeformed bubble ζ = z c , andthe spring attached to the tip is undeflected. Hence ingeneral the deflection is δ t = ζ − z c , and force exertedby the tip on the bubble is F t = − k t δ t = − k t [ ζ − z c ] , (2.34)where k t is the tip spring constant.Now z = z c ( r ) + ε ( r ), or ζ = ζ c ( r ) + ε ( r ) . (2.35)The amount of bubble deformation at contact is ε ( r ) = − k − F, (2.36)where F is the force exerted by the bubble. The bubblespring constant is given by the profile equation evaluatedat r , k bub = − πγ (
12 ln 1 − p − x p − x + 1 p − x − C ) − . (2.37)This depends upon the two pinning radii. Obviously R c > r > r .Figure 3 shows the exact effective spring constant ofthe bubble, ∆ F/ ∆ z ( r ) obtained from the numerical in-tegration of the profile equation, Eq. (2.26), normalizedby the analytic expression obtained from the expansionto linear order in the force, Eq. (2.37). The expansionis valid when | F | ≪ πγ ( s ) r /R c . For the conditions in k bub / g x FIG. 4: Bubble linear spring constant as a function of tipcontact radius x ≡ r /R c for a substrate radius x ≡ r /R c of 0.9 (solid curve), 0.7 (dashed curve), and 0.5 (dotted curve).In each case, the upper limit is x max1 = [1 − p − x ] tan α ,with α = 10 ◦ . Fig. 3, the right hand side is 0.09, 0.38, and 0.85 nN for r = 10, 20, and 30 nm, respectively. One can indeed seein the figure that the bubble behaves linearly to withinabout 20% of the exact value when the loads lie withinthis bound.Figure 4 shows the ratio of the bubble linear springconstant to the surface tension. It can be seen that thevariation is rather weak over the practical range, with thebubble spring constant being not more than a factor oftwo larger than the surface tension. In general the bubblespring constant is larger than the surface tension unlessthe contact radius is very small. Clearly, in the absenceof specific information about the size of the pinning radii,one will not go too far wrong in taking the liquid-vaporsurface tension of the supersaturated interface to be 0.5–1 times the measured bubble spring constant.In the linear regime, the bubble and tip act as twosprings in series. To see this explicitly, consider a changein the position of the substrate, ∆ ζ s , at constant r (and r ). Since ∆ r = 0, the change in contact position mustequal the change in tip position, ∆ ζ = ∆ ζ . In the staticsituation ∆ F t = − ∆ F , and so one has k t ∆ ζ = − k bub ∆ ε ( r )= − k bub [∆ ζ − ∆ ζ c ( r )]= − k bub [∆ ζ − ∆ ζ s ] . (2.38)This implies that [ k t + k bub ] ∆ ζ = k bub ∆ ζ s , or∆ ζ ∆ ζ s = k bub k t + k bub . (2.39)This ratio is less than unity, whereas for a hard surface itwould be unity. A measurement of the slope of the deflec-tion versus the drive distance allows the spring constantof the interface to be determined. If the contact radiiare known, then the surface tension of the bubble can beestimated. r r FIG. 5: Hemispherical bubble penetrated and deformed by aconical hydrophobic tip (blunt radius r ) upon which it slips. Alternatively, in terms of the separation between thetip and substrate, z ≡ ζ − ζ s , one has∆ ζ ∆ z = ∆ ζ ∆ ζ − ∆ ζ s = 11 − ∆ ζ s / ∆ ζ = 11 − k t + k bub k bub = − k bub k t . (2.40)In words, the slope of the deflection versus separationcurve equals the negative of the ratio of the bubble andcantilever spring constants. III. DEFORMED BUBBLE: PRICK SLIP
For the case of slip on the tip, the liquid-vapor-solidcontact circle is not pinned at r (see Fig. 5). One nowhas also to maximize the total entropy with respect to r , taking into account the difference in solid surface en-ergies, ∆ γ ≡ γ sg − γ sl . This is negative for a hydrophobictip.For a macroscopic bubble or droplet on a planar sub-strate made of the same material as the tip, the equilib-rium condition is that the contact angle measured in theliquid phase satisfies γ cos θ t = ∆ γ . This does not hold inthe present case of a conical tip penetrating the bubble.For the present case of slip, the total entropy is S tot [ z ] = S b ( N, V, T ) − γT A [ z ] − pT V [ z ] + µT N − ∆ γT A t . (3.1)The surface area of the blunt conical tip inside the bubbleis A t = πr sin α − πr sin α + πr . (3.2)The formulae for the volume V [ z ] and area A [ z ] weregiven above and depend upon r . Also recall that r − r = [ z − z ] tan α .Differentiating with respect to the profile (at constant r ) gives the Eular-Lagrange equation for the optimumprofile, as given above. Differentiating with respect to r , one has d r = d z tan α , which means that δz ( r ) =d r / tan α , and so the extra boundary term in the profilederivative of the area appears, as in § II B 1. Also, d z =0. In view of this one has ∂S tot ∂r (cid:12)(cid:12)(cid:12)(cid:12) z ( r ) (3.3)= (cid:20) ∆ p T δV [ z ] δz ( r ) − γT δA [ z ] δz ( r ) (cid:21) z ( r ) ∂z ( r ) ∂r + (cid:20) πr tan α + 2 πr (cid:16) z − r tan α (cid:17)(cid:21) ∆ p T + " πr p z ′ ( r ) + 2 πr z ′ ( r ) p z ′ ( r ) tan α γT − πr ∆ γT sin α = 2 πr T (" z R c + p z ′ ( r ) + z ′ ( r ) / tan α p z ′ ( r ) γ − ∆ γ sin α ) . Setting this to zero, the trivial solution is r = 0. Thenon-trivial solution gives an equation that the profileslope at the optimum radius must satisfy,∆ γγ sin α = 2 z R c + p z ′ ( r ) + z ′ ( r ) / tan α p z ′ ( r ) . (3.4)This replaces the planar contact angle condition.In the limit α → γγ = z ′ ( r ) p z ′ ( r ) = cos θ t , (3.5)which is the expected contact angle condition for a cylin-drical tip.The expression for the force ought to be unchanged. Explicitly the force exerted by the bubble on the tip is F = T d S tot d z = T (Z r r d r δS tot ([ z ] , r , z ) δz ( r ) (cid:12)(cid:12)(cid:12)(cid:12) [ z ] ,r d z ( r )d z + ∂S tot ([ z ] , r , z ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) [ z ] ,r d r d z + ∂S tot ([ z ] , r , z ) ∂z (cid:12)(cid:12)(cid:12)(cid:12) [ z ] ,r ) = T ∂S tot ([ z ] , r , z ) ∂z (cid:12)(cid:12)(cid:12)(cid:12) [ z ] ,z = πr ∆ p + 2 πr γz ′ ( r ) p z ′ ( r ) . (3.6)Hence the expression for the profile is unchanged from thestick case, although of course for a non-stick hydrophobictip, F <
5. Algorithms
Recall the equations of for the undeformed profile,Eq. (2.5) et seq.
Also recall that in the linear approxima-tion the deformation ε ( r ) ≡ z ( r ) − z c ( r ), and its deriva-tive ε ′ ( r ) ≡ z ′ ( r ) − z ′ c ( r ), are linearly proportional to theforce, Eqs (2.30) and (2.31). In view of these one cansuccinctly write z ( r ) = z c ( r ) − k bub ( r ) − F, (3.7)and z ′ ( r ) = z ′ c ( r ) + q bub ( r ) F. (3.8)Write k ≡ k bub ( r ) and q ≡ q bub ( r ).To linear order the optimum contact radius satisfies(recall that r − r = [ z ( r ) − z ] tan α )0 = " z R c + p z ′ ( r ) + z ′ ( r ) / tan α p z ′ ( r ) γ − ∆ γ sin α = " z c ( r ) R c + p z ′ c ( r ) + z ′ c ( r ) / tan α p z ′ c ( r ) γ − ∆ γ sin α − γFk R c + " z ′ c ( r ) tan α p z ′ c ( r ) + 1 p z ′ c ( r ) − z ′ c ( r ) [1 + z ′ c ( r ) ] / (cid:21) γq tan α F (3.9)Rearranging this gives explicitly F ( r ).Let the equilibrium curve be F ( r ), ζ s ( r ). The deflec-tion is δ t = z + ζ s − z c = − F t /k t = F/k t . The equilib-rium curve in linear approximation can be generated asfollows: D e f l ec t i o n ( n m ) -0.50.51.5 0 10 20 30 40 50 60 D e f l ec t i o n ( n m ) Separation (nm)
FIG. 6: Equilibrium (slip) deflection versus separation curvesfor a tip penetrating a nanobubble for different tip radii.These are the exact theory for, from bottom to top at contact,blunt tip radii of r = 0, 20, 30, 40, and 50 nm. The dottedline is a guide to the eye. The tip has α = 10 ◦ , k t = 0 .
35 N/m,and ∆ γ = 0 N/m or θ t = 90 ◦ . All other parameters as inFig. 2. • choose r • calculate F ( r ), and z = z ( r ) + [ r − r ] / tan α • calculate ζ s from F ( r ) = − F t = k t [ z + ζ s − z c ] • plot deflection, δ t = F ( r ) /k t versus separation, z .For the exact, non-linear calculations, one has to spec-ify the load F and calculate the profile z ( r ; F ) by numer-ical integration from ( r , r wherethe profile has the equilibrium contact angle, Eq. (3.4).In the cases where there are two solutions one choosesthe one based on continuity. From the value of z ( r ),one obtains z and ζ s as in the linear case. One thenchooses a new load and repeats the process. For the caseof the pinned tip contact line, for each F one insteadterminates the profile at the fixed value of r .
6. Results
Figure 6 shows several equilibrium deflection versusseparation curves. Equilibrium here and below mean thatthe contact circle on the tip is free to move. Hence theoptimum contact angle that maximizes the entropy isestablished at each separation. For brevity this is alsocalled slip. The data in the figure is obtained with theexact theory. Results obtained with the linear theory areentirely obscured by the exact curves on the scale of thefigure.In the case of Fig. 6, the tip has been taken to beindifferent to water, ∆ γ = 0 N/m or θ t = 90 ◦ , and forthe most part the force is repulsive. This correspondsto a positive cantilever deflection and to a compressed -202 D e f l ec t i o n ( n m ) -6-4 0 10 20 30 40 50 60 70 80 D e f l ec t i o n ( n m ) Separation (nm)
FIG. 7: Equilibrium (slip) deflection versus separation fora blunt tip penetrating a nanobubble for different tip surfaceenergies. The solid curves are the exact theory and the dashedcurves are the linear theory (obscured in the top three cases),with r = 20 nm and, from bottom to top, ∆ γ = − . θ t = 98 ◦ ), − . θ t = 95 ◦ ), − . θ t = 92 ◦ ),0 mN/m ( θ t = 90 ◦ ), and +2 . θ t = 88 ◦ ). No exactsolution was found for the lowest curve. The dash-dottedcurve and the partially obscured dash-double dotted curveare the exact and linear results respectively for r = 50 nm,and ∆ γ = − . θ t = 95 ◦ ). All other parameters as inFig. 6. bubble, as in Fig. 1 and in the lower half of Fig. 2. Theexception is for for the infinitely sharp tip, r = 0 nm,which shows a weak attraction.It is noticeable that the force curves are almost straightlines, and that their magnitude increases with increasingtip radius. The predominant reason that the force in-creases with decreasing separation is the cone half angle,which means that the contact radius increases as the tippenetrates further into the bubble with decreasing sep-aration. The main reason that the force becomes morerepulsive with increasing end radius r is that the repul-sive pressure contribution is proportional to r whereasthe often attractive surface tension contribution containsa factor r .Figure 7 explores the effect of the tip surface energydifference on the force curves. The conversion of the sur-face energy difference to a macroscopic tip contact an-gle uses the Young equation, the saturation value of thesurface tension, γ † = 0 .
072 N/m, and assumes that thedifference in surface energies is unchanged by the level ofsupersaturation of the solution, ∆ γ = γ † cos θ t .It can be seen that as the surface energy differencebecomes more negative, the force at a given separationbecomes increasingly attractive. The penetrated bubbleis extended from its undeformed shape (see Fig. 5 andthe upper half of Fig. 2). The slope of the almost lineardeflection versus separation curves changes from nega-tive to positive for ∆ γ < ∼ − . θ t > ∼ ◦ ), close towhere the surface energy difference changes sign, and itbecomes increasingly positive as the surface energy dif-ference becomes increasingly negative.The linear theory becomes increasingly less accurate asthe surface energy difference becomes more negative. In-deed, whilst the linear theory produced a solution curvefor ∆ γ ≤ − . r = 20 nm, no exact solu-tion was found (essentially because the bubble rupturedat this contact radius).Figure 7 also presents the case of r = 50 nm and∆ γ = − . θ t = 95 ◦ ). Compared to the samesurface energy but for r = 20 nm, one sees that theforce is less attractive and as well the slope has decreased,changing from positive to negative. As mentioned above,increasing the contact radius increases the repulsive pres-sure contribution to the force more than the attractivesurface tension contribution. One can conclude that anegative slope is the signature of pressure dominance(large contact radius, small magnitude or positive surfaceenergy difference) whereas a positive slope is the signa-ture of surface energy dominance, (small contact radius,large negative surface energy difference).In Figs 6 and 7 it can be seen that there is a discontin-uous jump from zero deflection prior to contact with theundeformed bubble to a non-zero deflection immediatelyafter contact. This is due to the fact that non-contactforces are neglected in the present model and also to thefact that the end of the tip has been taken to be a disc(ie. perfectly blunt, planar). In the present calculationsliquid-vapor interface contact with the flattened end hasbeen excluded. The size of this jump increases with thetip end radius r . Obviously this is an idealized modelof the actual tip of the tip, which is actually neither per-fectly sharp nor perfectly blunt, and in practice theremay be a smooth transition rather than a jump. Thecantilever manufacturer typically quotes a tip radius 20–60 nm. Also of course the AFM cantilever tip is generallyin the form of a square pyramid rather than the presentright circular cone.Figure 8 shows both a slip trajectory and a stick-sliptrajectory for a blunt tip, r = 20 nm. The stick brancheswere chosen more or less randomly, with one eye on aes-thetics, one eye on fundamental considerations, and oneeye on experimental data. There is no fundamental rea-son that when the contact line gives way it should jumpto, and immediately stick at, the equilibrium position. Itcould stick prior to reaching the equilibrium position, orit could slip along the equilibrium curve after the jump.One could perhaps argue for a yield stress such that thecontact line always slipped when the excess force per unitcontact line reached a certain value, but this has not beendone here.The first tip contact line stick has been taken to occurat r = 26 . r =28 . -3-2-10 D e f l ec t i o n ( n m ) -6-5-4 0 10 20 30 40 50 60 70 80 90 D e f l ec t i o n ( n m ) Separation (nm)
FIG. 8: Deflection versus separation for a tip penetratinga nanobubble. The solid curve is the equilibrium slip caseusing the exact theory, and the parallel dashed curve uses thelinear theory. The dash-dotted curve is a stick-slip case usingthe exact theory, with the pinned contact radii being r =26.4, 28.9, and 31.6 nm (chosen more or less arbitrarily). Theapproach curve overlaps the initial part of the retraction curveon the final stick region. The dotted arrows show cantileverjumps. The final jump out is at the limit of bridging bubblestability for r = 31 . r = 20 nm,and the difference in tip surface energies is ∆ γ = − . θ t = 95 ◦ ), with all other parameters Fig. 6. erage slope of the entire branch is -0.95. The linear pre-diction for the slope at this contact radius is -0.143.The third tip contact line stick occurs at r = 31 . r = 31 . F = − IV. EXPERIMENTA. Limitations of Theory
Before analyzing any experimental data, it is worthenumerating the limitations of the present theory.First, the theory models the tip as a right circular cone,whereas in reality an AFM tip is a rectangular pyramid.Second, it models the tip of the tip as perfectly blunt, adisc of radius r , whereas in reality the tip may be curvedfrom wear. Third, it is assumed that the cross-section ofthe nanobubble is perfectly circular, which need not bethe case if substrate heterogeneity determines where thecontact line is pinned. Fourth, it is assumed that the sub-strate contact line does not alter during the penetrationof the nanobubble by the tip.Fifth, it is assumed that the tip penetrates thenanobubble at the apex, whereas in reality it can pen-etrate off-axis, either by design or by accident. Sixth, itis assumed that the tip is oriented normal to the sub-strate, whereas in reality the cantilever and tip are tiltedat about 11 ◦ in the axial plane. Seventh, it is assumedthat the air inside the nanobubble remains in diffusiveequilibrium with that in the solution during the forcemeasurement (constant chemical potential), whereas inreality the measurement might be rapid enough for con-stant number to hold instead.The fifth and sixth points mean that the predicted nor-mal force on the cantilever has an error that increaseswith displacement from the central axis of the nanobub-ble, that the displacement from the central axis varieswith the separation and with the deflection of the can-tilever during a force measurement (due to the tilt),and that there is a torque on the cantilever due to theasymmetric forces on the tip when it passes through thenanobubble interface off-axis. This nanobubble torque isin addition to the torque that acts in all AFM force mea-surements on the tip in contact with the hard substratedue to the normal and lateral (friction) forces, which areimplicitly included in the photo-diode calibration. The limitation summarized in the fifth point can besubstantially alleviated by ensuring that the separationat which first contact with the nanobubble occurs is equalto the measured height of the undeformed nanobubble.One is aided in this by the fact that the nanobubble pro-file is horizontal near the apex. In this case one can beconfident that, at least for small cantilever deflectionsand separations close to initial contact, the tip is pene-trating the nanobubble close to the apex, and the cali-bration factor is correct. In this regime the present cal-culation of the nanobubble normal force and the neglectof any nanobubble lateral forces or torques, ought to beaccurate. The consequences of the seventh point are dis-cussed below on p. 11.In view of these limitations, one ought not to expectthe present theory to be able to quantitatively describeevery aspect of an individual nanobubble force measure-ment. The main goal in the first instance is to establish that the nanobubble surface tension is less than the usualair-water surface tension, and if possible to quantify re-liably its value for a given solution.To this end the following protocol was adopted. Pri-mary emphasis was placed on the first pinned region ofthe measured separation-deflection curve, since this is theone that can be guaranteed closest to the apex. Addi-tional pinned regions were used to confirm the valuesdeduced from the first one.Since the surface tension that is required to fit a givenslope decreases with increasing value of r > ∼ r , one canestablish an upper bound for the surface tension by spec-ifying the lowest realistic value of r . In view of the spec-ification that a new tip has radius in the range 20–60 nm,fixing for example r = 20 nm should give an upper limiton the surface tension.Further, as is discussed below on p. 11, for a givensurface tension and pinned contact radius r , the slopecalculated at constant chemical potential is less in magni-tude than the slope calculated at constant number. Thismeans that the surface tensions obtained below at con-stant chemical potential are larger than those that wouldbe required to fit the slopes at constant number. Againone can be confident that the surface tensions obtainedhere are an upper bound on those for actual nanobubbles.Since the slope of the pinned regions equals the neg-ative of the ratio of the nanobubble spring constant tothe cantilever spring constant, the surface tension cannow be determined using the linear theory. The accu-racy of this can be checked against the exact theory.From the surface tension and the nanobubble curvatureradius determined by tapping mode imaging, the super-saturation ratio is now determined, Eq. (2.1). Usinga linear model for the supersaturated surface tension, γ ( s ) = ( s ‡ − s ) γ † / ( s ‡ − where γ † = 0 .
072 N/mis the saturated surface tension, the spinodal supersatu-ration ratio s ‡ can now be determined. B. Nanobubble 1
Figure 9 shows AFM measurements of the force ona cantilever tip due to a single nanobubble. What isplotted is the positional deflection of the cantilever; toobtain the force, multiply by the cantilever spring con-stant, k t = 0 .
35 N/m. The first measurably significantdeflection occurs at a separation of z = 32 . z c = 33 nm. Thissuggests that pre-contact forces (van der Waals, electricdouble layer) are negligible. It also confirms that themeasurement was performed in the central region of thenanobubble close to the apex.The nanobubble profile obtained from the image (notshown) has a contact radius of r = 108 nm, which, withits height, corresponds to a contact angle of 146 ◦ . Thefact that this contact angle is substantially higher thanthe contact angle of a macroscopic water drop on HOPG,1 -505 D e f l ec t i o n ( n m ) -.27-.29-15-10 0 10 20 30 40 50 D e f l ec t i o n ( n m ) Separation (nm)
FIG. 9: Cantilever deflection versus separation for a SiNtip penetrating a nanobubble on an HOPG substrate on ap-proach (triangles) and retraction (crosses). The cantileverhas spring constant k t = 0 .
35 N/m and conical half-angle α = 10 ◦ . The undeformed nanobubble has measured height z c = 33 nm and substrate contact radius r = 108 nm, corre-sponding to a curvature radius R c = 192 nm and a contactangle of 146 ◦ . The dotted lines and arrows are guides to theeye, with the adjacent number giving the slope. The full anddashed curves are respectively the calculated exact and linearequilibrium deflection (tip contact line slip) with r = 10 nm, γ = 0 .
040 N/m, s = 5 .
16, and, ∆ γ = − . θ t = 98 ◦ ). The dash-double dotted line ( r = 10 nm, r = 6 nm, obscured), and the dash-dotted curve ( r = 20 nm, r = 10 nm), are calculated exact deflections with the tip con-tact line pinned, using γ = 0 .
044 N/m, s = 5 . ◦ , is strong evidence that the nanobubble con-tact rim is pinned. In the experimental data just after first nanobubblecontact, the positively sloped region, followed by a briefplateau, followed by a small jump to the base of the firstmarked linear region, are all due to the initial spreadingof the nanobubble on the tip of the tip and up its sides.This region is not well-modeled by the present geometryof the tip of the tip as a perfectly planar circular disc.The fact that the deflection is negative in this regionindicates that it is favorable for the tip to penetrate thenanobubble, which is to say that the SiNi tip must behydrophobic, or possibly barely hydrophilic.The two linear regions with labeled slopes evident inthe experimental deflection data on approach confirmthat the bubble can have a pinned contact line and be-have as a Hookean spring. Since the cone half angle issmall, α = 10 ◦ , to leading order one can take the contactradius used in fitting the slopes to be the same as theradius of the perfectly blunt tip, r ≈ r . The manufac-turer’s quoted radius of the tip, 20–60 nm, which mightrefer to either the tip’s width or else its radius of curva-ture, can be assumed to be of the same order as r in thepresent simple model.Choosing a contact radius at the lower end of the re-alistic values, r = 10 nm, a value of γ = 0 .
040 N/mgives k b /k t = 0 .
27, which is the negative of the measured slope of the first linear region. Using Eq. (2.1), this sur-face tension requires a supersaturation ratio of s = 5 . R c = 193 nm, equal to thatdeduced from the tapping mode images of this partic-ular nanobubble. Conversely, choosing the upper limit r = 50 nm, the fitted surface tension is γ = 0 .
015 N/mand s = 2 .
58. Hence even in the most pessimistic caseof smallest contact radius one can see that the nanobub-ble surface tension, γ = 0 .
040 N/m almost a factor oftwo smaller than the surface tension of the saturated air-water interface, γ = 0 .
072 N/m, and that the solution issubstantially supersaturated with air, s = 5.The just quoted slopes were obtained with the lineartheory. Applying the exact non-linear theory with thecontact line again pinned at r = 10 nm, the tangentat the start of the dash-double dotted curve in Fig. 9,gives the required slope − .
27 using γ = 0 .
044 N/m and s = 5 .
55. In this case using r = 6 nm shifts the curvelaterally to coincide with the measured data. Obviouslyusing larger values of r will require smaller values of γ ( s )to fit the slope. There is very little curvature evident inthe non-linear curve. The 10% difference in the surfacetension between the exact and the linear fits means thatthe linear theory provides an acceptable estimate of thesurface tension from the slope that is both analytic andreliable.The slope of the second linear region in Fig. 9, − . γ = 0 .
043 N/m using r = 10 nm using thelinear theory. Alternatively, the change in contact posi-tion of the nanobubble on the tip may be approximatedas the change in separation at the base of the two lin-ear regions, ∆ z ≈ ∆ z = 11 nm, assuming that thebubble profile is essentially the same in the two cases,which it would be if the contact line were mobile priorto the start of the pinned regions. The change in con-tact radius is ∆ r = ∆ z tan α ≈ γ = 0 . r = 10 nm,and a slope of -0.29 using r = 12 nm. In the case of r = 50 nm and r = 52 nm, the two slopes are given bya single surface tension with a slightly worse variance of2%. This tends to suggest that the smaller contact radiusis more applicable, but this is by no means conclusive. Inany case, that a single surface tension combined with thegeometry of the tip fits the two slopes supports the modeland the value of the surface tension.Here and below the calculations are performed at con-stant chemical potential, which is computationally conve-nient. Although there is strong evidence (see below) thatthe nanobubble is in diffusive equilibrium with the solu-tion over the time of the series of force measurements, itis unclear whether it is best to model each force measure-ment as at constant chemical potential or as at constantnumber. For the purposes of comparison, some exactcalculations have been carried out at constant number.Using a surface tension of γ ( s ) = 0 .
044 N/m, at a pinnedradius of r = 10 nm the tangent at zero force at constantchemical potential is -0.30, compared to -0.41 at constant2number. Using instead r = 20 nm and the same surfacetension, the tangent at zero force at constant chemicalpotential is -0.40, compared to -0.62 at constant num-ber. One sees that the slope has a higher magnitude atconstant number, and that it increases relatively morerapidly with contact radius. Hence one would requiresmaller surface tensions to fit the measured slopes if oneused constant number. The calculations here and beloware at constant chemical potential, and so the surface ten-sions obtained represent an upper bound on the actualnanobubble surface tension.The two almost horizontal curves in Fig. 9 are the cal-culated exact and linear equilibrium curves, which as-sume that the contact line is mobile on the tip. The goodagreement between the linear and the exact calculationsis somewhat better than that for the predicted effectivebubble spring constant. Both equilibrium calculationsuse r = 10 nm, γ = 0 .
04 N/m, and s = 5 .
16. In ad-dition a surface energy difference of ∆ γ = − . θ t = 98 ◦ . Thisis slightly hydrophobic. The criterion for the fit, whichwas done by eye, was that the curve should pass close tothe base of the two linear regions. (For reasons that arediscussed below, the end point of the final jump was notincluded in the fit.) Since the cantilever jumps to thesebases, the nanobubble at contact must also be movingalong the tip, and so it can probably be assumed thatthe contact position is the equilibrium one. Of coursethe contact line may become pinned at the end of thejump prior to achieving its equilibrium position, so thisfit may underestimate the magnitude of the surface en-ergy difference. Also, using a larger contact radius wouldrequire a smaller in magnitude value for the fit. Fortu-nately, the surface tension obtained by fitting the slopesof the linear regions is not affected by the tip solid surfaceenergies.From the fitted value of the surface tension at r =10 nm, γ = 0 .
040 N/m, and the measured nanobubblecurvature radius, R c = 193 nm, which is equal to thecritical radius, Eq. (2.1), the supersaturation ratio canbe deduced to be s = 5 .
2. The linear model for the su-persaturated surface tension is γ ( s ) = ( s ‡ − s ) γ † / ( s ‡ − γ † = 0 .
072 N/m is the saturated surface tension,and s ‡ is the spinodal saturation ratio. This has beenshown to fit the available computer simulation data rea-sonably accurately. These computer simulationsgive s ‡ ≈ γ (5 .
2) = 0 .
040 N/m,gives the spinodal supersaturation ratio s ‡ = 10 .
4. Al-ternatively, at r = 50 nm, the linearly fitted supersat-urated surface tension γ = 0 .
015 N/m requires s = 2 . s ‡ = 3 . r = 20 nm. The value of the contact ra-dius was chosen so that the calculated curve fitted by eye the measured data at the end of the retractionbranch. This calculation used the non-linear fitted value γ = 0 .
044 N/m and s = 5 .
55, and also r = 10 . − .
27 ( r = 10 nm and r = 6 . r it could doubtless be made even bet-ter. It is also undoubtable that larger values of r and r and smaller values of γ ( s ) could also fit both the slope ofthe pinned regions on approach and the flattened regionon retraction. The conclusion that one can draw is thatfor separations z > ∼
20 nm the retraction data can bedescribed by the pinned nanobubble model using param-eters consistent with what was deduced from the slopesof the pinned approach data.For separations z < ∼
20 nm on retraction, and z < ∼ The origin of this particu-lar behavior is unclear. Because of the coincidence of ap-proach and retraction here, this is clearly an equilibrium,non-dissipative phenomenon. Calculations show that itis not due to elastic deformation of the substrate (notshown).
It might be due to torque on the cantilever,although measurements across the nanobubble indicatethat this is in general negligible. That the curve is muchsteeper at F = 0 than the clearly pinned regions suggeststhat it is not due to pinning of the contact line, unlessthe pinned contact radius had increased very substan-tially. The apparently contiguous flat region r > ∼
20 nmon retraction is similar to the non-linear calculations ofthe force due to pinning of a highly extended nanobub-ble with small contact radius. Obviously whatever theorigin of this behavior, it could be simply additive to theforce due to the pinned (or slipping) contact line sincethe nanobubble force is always present. Because of theuncertainty as to the origin of this force close to contactit has been neglected in fitting the nanobubble.The measurements in Fig. 9 were part of a sequenceof twelve successive force measurements across this par-ticular nanobubble (not shown). The number, position,and extent of the linear regions could differ between forcemeasurements, presumably because contact line stick andslip are stochastic events, but the slopes were unchanged.This suggests that torque on the cantilever due to off-apex penetration has negligible effect. Tapping modeimages before and after the sequence of force measure-ments show that the nanobubble itself was unchanged insize and shape by the force measurements. This is strongevidence that the nanobubble is thermodynamically sta-ble, and that even penetrating it a dozen times with thecantilever tip did not destroy or alter it. It is also evi-dence that the nanobubble is pinned at it contact rim.In several other series of measurements, up to a hundredforce measurements were performed on a single nanobub-ble, interspersed with several AFM tapping mode images,3 -3-2-10123 D e f l ec t i o n ( n m ) -.50-.57-7-6-5-4-3 0 5 10 15 20 D e f l ec t i o n ( n m ) Separation (nm) -.66
FIG. 10: Measured cantilever deflection versus separation fora nanobubble (curves and lines as in preceding figure; dif-ferent nanobubble, cantilever, and solution). The cantileverhas spring constant k t = 0 .
24 N/m, and conical half-angle α = 10 ◦ . The undeformed nanobubble has measured height z c = 9 . r = 82 . R c = 359 nm and a contact an-gle of 167 ◦ . The calculated equilibrium exact (solid) and lin-ear (dashed, obscured) curve use r = 10 nm, γ = 0 .
041 N/m, s = 3 .
3, and ∆ γ = +1 . θ t = 89 ◦ ). and no significant change in the nanobubble was observedin any case. C. Nanobubble 2
Figure 10 shows results for another nanobubble in adifferent solution, with stick-slip behavior evident. As-suming an initial tip contact radius of r = 10 nm, themeasured slope − .
50 corresponds to a surface tension of γ ( s ) = 0 .
041 N/m (linear approximation). Using instead r = 20 nm gives γ ( s ) = 0 .
028 N/m. Larger contact radiirequire even smaller surface tension to yield this slope.Exact calculations differ by less than 1% from the linearresults for the bubble spring constant in this regime.The second pinned region with slope -0.57 correspondsto γ ( s ) = 0 .
047 N/m using r = 10 nm, and to γ ( s ) =0 .
032 N/m using r = 20 nm. The third pinned regionwith slope -0.66 corresponds to γ ( s ) = 0 .
054 N/m using r = 10 nm, and to γ ( s ) = 0 .
037 N/m using r = 20 nm.From the point at which the pinned regions extrapo-late to zero deflection, one can deduce the value of thepinned radius r for a specified value of the tip radius r . However, it is not possible to find a single value of r which yields a single surface tension when all threeslopes are fitted. For example, fixing r = 10 nm, onefinds that the three slopes -0.50, -0.57, and -0.66 corre-spond to r = 10.9, 11.1, and 11.7 nm, and to γ ( s ) =0.039, 0.045, and 0.050 N/m, respectively. There is noth-ing wrong with r increasing with each successive pin-ning event as the tip penetrates the nanobubble, but onewould have hoped for a single surface tension. The best D e f l ec t i o n ( n m ) -.29-101 0 5 10 15 20 25 D e f l ec t i o n ( n m ) Separation (nm) -.28
FIG. 11: Measured cantilever deflection versus separation fora nanobubble (curves and lines as in preceding figure; dif-ferent cantilever, solution, and nanobubble). The cantileverhas spring constant k t = 0 .
35 N/m, and conical half-angle α = 10 ◦ . The undeformed nanobubble has measured height z c = 19 . r = 205 nm, cor-responding to a curvature radius R c = 1087 nm and a contactangle of 169 ◦ . The dashed curve is equilibrium (slip) lin-ear theory with r = 20 nm, γ = 0 .
037 N/m, s = 1 .
68, and∆ γ = +3 . θ t = 87 ◦ ). that can be done (ie. minimizing the sum of the rela-tive standard deviations in surface tension and in tip ra-dius), yields r = 12 . ± .
001 nm and γ ( s ) =0.036,0.041, and 0.046 N/m, respectively. Possibly doing thecalculations at constant number rather than the presentconstant chemical potential might yield more consistentresults (see p. 11). Or possibly the problem is relatedto the unknown origin of the steep hook at small sep-arations discussed on p. 12. In any case, the preferredvalue of surface tension is the one taken from the veryfirst pinned region, since this lies closest to the tip of thetip of the cantilever and to zero force.The surface tension obtained using r = 10 nm, γ ( s ) =0 .
041 N/m, and the nanobubble radius of curvature R c =359 nm correspond to a supersaturation value of s = 3 . s ‡ = 6 .
3. Using instead r = 20 nm, γ ( s ) = 0 .
028 N/m,correspond to a supersaturation value of s = 2 .
6. and aspinodal supersaturation ratio of s ‡ = 3 . r = 10 nm, γ ( s ) = 0 .
041 N/m, and s = 3 .
3. The value ofthe surface energy difference, ∆ γ = +1 . θ t = 89 ◦ ) was chosen so that the curves passedthrough the base of the first pinned region. The exactand the linear calculations are almost indistinguishable. D. Nanobubbles 3 and 4
Figure 11 shows yet another nanobubble measurement.The AFM fluid cell was flushed with ethanol and then4water which means that there was likely exothermicheating. The undeformed nanobubble has apex height z c = 19 . z c = 21 . r = 215 nm, which correspond to a curvature radius R c = 1081 nm. Again this is unambiguous evidence thatthe nanobubble is thermodynamically stable.The two linear regions measured in the figure, one eachon extension and retraction, may be attributed to stick.Assuming a contact radius of r = 20 nm, the slope − . γ ( s ) = 0 .
037 N/m. Alternatively for thisslope, r = 10 nm corresponds to γ ( s ) = 0 .
048 N/m. and r = 50 nm γ ( s ) = 0 .
022 N/m. The exact and linearbubble spring constants agree to better than 0.05% inthis regime.The value γ ( s ) = 0 .
037 N/m, and the nanobubble ra-dius of curvature R c = 1070 nm correspond to a super-saturation value of s = 1 .
7. Using this in the linearmodel for the supersaturated surface tension gives a spin-odal supersaturation ratio of s ‡ = 2 .
4. Alternatively, γ ( s ) = 0 .
048 N/m corresponds to s = 1 . s ‡ = 3 . r = 20 nm, these data are well-fittedusing γ ( s ) = 0 .
037 N/m, s = 1 .
7, and ∆ γ = +3 . θ t = 87 ◦ . The va-lidity of the fit is supported by the coincidence of thejump-out separation and the end of the stability of theextended nanobubble. It can be mentioned that a vir-tually identical equilibrium slip curve can be obtainedfor r = 50 nm, with γ ( s ) = 0 .
022 N/m, s = 1 .
41, andwith ∆ γ = +2 . θ t = 88 . ◦ . The equilibrium data can also be fitted by r = 10 nm, with γ ( s ) = 0 .
048 N/m, s = 1 .
9, and with∆ γ = +2 . θ t = 88 ◦ ).Figure 12 shows another nanobubble in the same solu-tion as Fig. 11. Despite the differences in height and con-tact radii, the two nanobubble have about the same ra-dius of curvature ( R c = 1087 nm there and 970 nm here).This is consistent with the level of supersaturation of thesolution being unchanged and with the nanobubbles be-ing in diffusive equilibrium. Likewise, one would expectthe surface tension to be unchanged, and this is confirmedby the fact that the slope of the linear pinned regions areabout the same ( − .
28 there and − .
29 here).Using a contact radius of r = 20 nm, the slope of thestick region in extension in Fig. 12 of − .
29 correspondsto a surface tension of γ = 0 .
035 N/m, a supersaturationratio of s = 1 .
72, and a spinodal supersaturation ratio of s ‡ = 2 .
4. The minimum of the extension curve touchesa calculated equilibrium slip curve for ∆ γ = +2 . θ t = 88 ◦ ). D e f l ec t i o n ( n m ) -.29-2-10 0 5 10 15 20 25 D e f l ec t i o n ( n m ) Separation (nm)
FIG. 12: Measured cantilever deflection versus separation fora nanobubble (same cantilever and solution as in the preced-ing figure; different nanobubble). The undeformed nanobub-ble has measured height z c = 15 . r = 174 nm, corresponding to a curvature ra-dius R c = 970 nm and a contact angle of 170 ◦ . The dashedcurve is equilibrium (slip) linear theory with r = 20 nm, γ ( s ) = 0 .
035 N/m, and s = 1 .
72, and ∆ γ = +2 . θ t = 88 ◦ ). Using instead a contact radius of r = 50 nm, the slopeof the stick region in extension in Fig. 12 of − .
29 cor-responds to γ ( s ) = 0 .
020 N/m, s = 1 . s ‡ = 1 .
58. Fit-ting the minimum of the extension curve gives ∆ γ =+1 . θ t = 89 ◦ ).Using instead a contact radius of r = 10 nm, the slopeof − .
29 corresponds to γ ( s ) = 0 .
046 N/m, s = 1 . s ‡ = 3 .
61. Fitting the minimum of the extension curvegives ∆ γ = +3 . θ t = 87 ◦ ).It should be mentioned that the slope of the stick re-gion was checked against that given by the exact theoryand the agreement was better than 1%. The origin of thenon-linearity and peak in the putative pinned region inFig. 12 (and in Fig. 11) is unclear, although one couldspeculate that the contact line might be moving with afinite velocity in these regions.The slight negative slope in the putative equilibriumcurve here in Fig. 12 is difficult to reproduce in the the-oretical calculations. V. CONCLUSION
The present paper gives analytic expressions for thenanobubble spring constant that allow its surface tensionto be obtained from the slope of the pinned regions ina force-separation AFM measurement. Expressions arealso given that allow the difference in tip surface energies(tip contact angle) to be obtained from an equilibriumpart of the force curve.The present fits to the experimental data do not giveenough information to pin down the value of the blunttip radius r . However, sensible results are generated5 TABLE I: Measured and Deduced Properties of Nanobubblesand Solutions ( r = 10 nm).Figure k t r z c R c γ ( s ) s s ‡ θ t (N/m) (nm) (nm) (nm) (N/m) (deg.)9 0.35 108 33 192 0.040 5.2 10.4 9810 0.24 82.5 9.6 359 0.041 3.3 6.3 8911 ∗ ∗ ∗ Same solution, different nanobubble. by assuming a tip and contact radius r = r = 10 nm(Table I). This is at the lower end of the range of a typicalAFM tapping mode cantilever tip, 20–60 nm. A highvalue of the radius, r = r = 50 nm gave quite lowvalues of the surface tension, supersaturation ratio, andspinodal supersaturation ratio.The most reliable data appears to be that of Fig. 9.In this case the two pinned regions both appear to beginfrom the equilibrium curve, which means that the changein contact radius can be found from the change in separa-tion of the starts. Hence one has three knowns (the twoslopes and the change in contact radius) and three un-knowns (the surface tension and the two contact radii).Solving this system yields the contact radii r = 9 . r ′ = 11 . γ ( s ) = 0 .
041 N/m, s = 5 . s ‡ = 10 .
8. Unfortunately the other figures do nothave pinned regions starting from the equilibrium curveand so the change in contact radius cannot be readilydeduced.The estimates of the surface energy difference andmacroscopic tip contact angle are based on the equilib-rium curves and are not so reliable. Little more can besaid than that the contact angle is close to 90 ◦ .Likewise the estimate of the value of the spinodal su-persaturation ratio has limited reliability because of thesimplicity of the linear supersaturated surface tensionmodel. It is nonetheless consoling that it comes out tobe of the same order as has been found in computer sim-ulations of a Lennard-Jones fluid. One of the purposes of this study was to establishexperimentally that the surface tension of nanobubbleswas less than that of saturated water. The surface ten-sion is obtained from the slope of the pinned regions anddoes not rely upon the surface energy difference nor thespinodal supersaturation ratio. The greatest uncertaintyconcerns the tip radius, and to this end it is better touse a low value, since this overestimates the surface ten-sion. (Doing the calculations at constant chemical po-tential rather than at constant number further overesti-mates the surface tension.) Hence the data in Table Ifor r = 10 nm, which give 0 . < ∼ γ ( s ) < ∼ .
05 N/m, aremost likely an upper bound on the nanobubble surfacetension. The nanobubble surface tension is substantiallyreduced from that of the saturated air water interface, γ † = 0 .
072 N/m.The solution supersaturation ratios are deduced to bein the range 1 . < ∼ s < ∼ . r = 10 nm. Thesevalues appear realistic given the fact that a 15 ◦ C changein temperature is enough to change the solubility of CO by a factor of two.It is clear that in order to reliably obtain the depen-dence of the surface tension on the supersaturation ratioone needs to know the contact radius to within a fewnanometers. In contrast, in order to prove that the solu-tion is supersaturated and that the surface tension is lessthan the saturated value one does not need to know thetip radius precisely.On this basis one can conclude that the present analy-sis of these experimental measurements on nanobubblesexplicitly confirm what is required by thermodynamics:for nanobubble equilibrium the solution must be super-saturated, and a supersaturated solution has a lower sur-face tension than a saturated solution. In future experimental studies, electron micrographyor AFM inverse imaging could be used to get an inde-pendent estimate of the cross-sectional radius of the tip.Also, attempts could be made to explicitly control or tomeasure the supersaturation of the solution.
Acknowledgement.
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