Geometry as a Catalyst: How Vapor Cavities Nucleate from Defects

Article pubs.acs.org/Langmuir

Geometry as a Catalyst: How Vapor Cavities Nucleate from Defects Alberto Giacomello,† Mauro Chinappi,‡ Simone Meloni,§,∥ and Carlo Massimo Casciola*,† †

Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Rome, Italy Center for Life Nano Science@Sapienza, Istituto Italiano di Tecnologia, Rome, Italy § CINECA Consortium, Rome, Italy ∥ Laboratory of Computational Chemistry and Biochemistry, Institute of Chemical Science and Engineering, École polytechnique fédérale de Lausanne, Lausanne, Switzerland ‡

S Supporting Information *

ABSTRACT: The onset of cavitation is strongly enhanced by the presence of rough surfaces or impurities in the liquid. Despite decades of research, the way the geometry of these defects promote the nucleation of bubbles and its effect on the kinetics of the process remains largely unclear. We present here a comprehensive explanation of the catalytic action that roughness elements exert on the nucleation process for both pure vapor cavities and gas ones. This approach highlights that nucleation may follow nontrivial paths connected with a sharp decrease of the free energy barriers as compared to flat surfaces. Furthermore, we demonstrate the existence of intermediate metastable states that break the nucleation process in multiple steps; these states correspond to what is commonly known as cavitation nuclei. A single dimensionless parameter, the nucleation number, is found to control this rich phenomenology. The devised theory allows one to quantify the effect of the geometry and hydrophobicity of surface asperities on nucleation. Within the same framework, it is possible to treat both vapor cavitation, which is relevant, e.g., for organic liquids, and gas-promoted cavitation, which is commonly encountered in water. The theory is shown to be valid from the nano- to the macroscale.

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also refs 17 and 18). In this thermodynamic theory, the vapor cavity is assumed to grow from the bulk liquid, to be spherical, and its radius to be the reaction coordinate. It is worth stressing that the Volmer theory focuses on vapor bubble nucleation. The case of systems containing dissolved gases is treated in other theories, such as the crevice model discussed below. The cost of forming the bubble interface from the bulk liquid gives rise to large free energy barriers for nucleation. Volmer theory, therefore, predicts that liquids can withstand extreme negative pressures before observing homogeneous nucleation from the bulk of a liquid. For instance, in water at ambient temperature, the homogeneous nucleation pressure exceeds −120 MPa: this limit has only recently been reached in carefully controlled experiments of ultrapure water inclusions in quartz.19,20

avitation is the formation of gas or vapor cavities within a liquid in a tensile state.1 Traditionally, in the field of hydraulic and marine engineering, cavitation is a much-feared phenomenon that leads to the rapid deterioration of fluid machinery and hydraulic structures.2,3 The abrupt collapse of cavitation bubbles is indeed accompanied by an intense increase of the local pressure and temperature4 that leads to surface reactions and physical modifications. In material science,5 the extreme conditions in these hot spots have been favorably exploited in the synthesis of biomaterials6,7 and nanostructured materials.8−10 In medicine, cavitation induced by ultrasounds shows promise as a noninvasive tool capable of in situ destruction of damaged tissues, occlusions, etc.;11−13 at the same time, cavitation may be exploited for drug delivery.14,15 A fundamental physical question underlies these diverse applications: Where and when is cavitation to be expected? A first theory of nucleation, known as classical nucleation theory, is due to Volmer and dates back to the late 1930s16 (see © 2013 American Chemical Society

Received: July 2, 2013 Revised: November 6, 2013 Published: November 11, 2013 14873

dx.doi.org/10.1021/la403733a | Langmuir 2013, 29, 14873−14884

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is encountered, revealing different mechanisms behind heterogeneous cavitation. In order to identify the nucleation path, we resort to the Continuum Rare Event Method (CREaM), explained below. CREaM offers a route to determine the nucleation path on surface defects having arbitrary geometry. We demonstrate its capabilities on three defects having different geometries: a rectangular groove and two conical crevices. These latter cases are reported for comparison with the available literature. The main results are derived for nucleation of vapor cavities on patterned surfaces. For this scenario, it is found that a single dimensionless parameter, the nucleation number Nnu, comparing defect size and critical nucleation radius, is crucial for rationalizing the nucleation mechanisms on a given surface geometry and chemistry. In the last section before the Conclusions, the route to include the effect of dissolved gas and contact angle hysteresis into our model is presented. For what concerns the spinodal pressure, we find substantial agreement with the crevice model;25 in addition, we are able to predict the free energy barriers, the coexistence pressure, and the nucleation path. Finally, in a manner analogous to what done for crystal nucleation28 and condensation29 via chemical patterning, we suggest that the present theory can be helpful in designing physically patterned surfaces for controlling liquid-to-vapor nucleation. In the present case, indeed, the geometry of surface patterns is the key to cavitation control that can be easily tuned with current microfabrication technologies.

However, in most real-world liquids, contaminant particles or solid walls are present promoting heterogeneous nucleation at these preferential sites. In addition, the presence of gas nuclei in the liquid may further dramatically reduce the nucleation pressure to few atmospheres.1 Volmer-like theories are not capable of explaining the 3 orders of magnitude separating the homogeneous cavitation threshold from the actual tensile strength of water; gas-promoted nucleation is the subject of other theories, the most successful of which is the crevice model.21−25 As will be shown next, both classes of theories cannot account for several qualitative aspects of nucleation from rough surfaces; most remarkably, they are not capable to capture the changes in the nucleation path induced even by geometrically simple surface corrugations. These drawbacks also limit the predictive capabilities of such theories on the rate of nucleation. Surface defects (especially when combined with hydrophobic coatings) and gas nuclei act as catalysts to accelerate nucleation events. Knowing how these catalysts act to change the nucleation rates may open new ways to control cavitation by engineering the surface microtexture and coatings. For instance, in material chemistry or ultrasound medicine, where the extreme conditions of cavitation are profitably used, this knowledge may allow to design nanoparticles or microstructured surfaces that promote cavitation. It would be therefore of great practical interest to identify the mechanism of catalysis of the three main elements that determine the heterogeneous nucleation threshold for real liquids, that is, surface geometry, surface chemistry, and the presence of gas. An important attempt to address this question is the aforementioned crevice model;21−25 here we refer to the version of Atchley and Prosperetti25 and its extensions.26 By assuming that the crucial element that determines cavitation is the unstable growth of the nuclei, this model allows one to accurately determine the spinodal pressure for cavitation from gas nuclei trapped in axisymmetric surface defects.27 The crevice model is based on a balance of the expanding (vapor and gas pressures) and collapsing (surface tension and liquid pressure) forces. For the very nature of its mechanical derivation, however, the crevice model cannot account for metastabilities involved in nucleation (away from the spinodal). In this paper, hoping to contribute to fill gaps in the theoretical description of nucleation, we develop a comprehensive theory for the formation of vapor/gas cavities from surface defects based on energy arguments, or, via statistical mechanics, on probabilistic arguments. In particular, we focus on the free energy barriers connected to the nucleation process, on its kinetics, and on the equilibrium conditions for vapor cavities entrapped in surface defects. These pieces of information are crucial, for instance, to determine whether heterogeneous nucleation is a deterministic processas typically occurs close to the spinodalor a stochastic one, with an Arrhenius-like kinetics. Furthermore, the present approach offers insights into the stability of vapor/gas nuclei trapped in surface defects that may be useful for developing models of cavitation for engineering contexts. We show that the crucial point to build an energy theory for heterogeneous nucleation is the nucleation path (or mechanism), that is, the shape of the nucleating bubble along the transition from the liquid phase to the unstable bubble. The nucleation path determines the free energy barriers, nucleation rates, and sites where cavitation will occur. Even on defects having very simple geometries, a multifaceted phenomenology

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RESULTS AND DISCUSSION Free Energy Profiles via CREaM. As anticipated, in the present approach, free energy profiles represent the key tool in understanding the mechanism and the kinetics of nucleation. The probabilistic interpretation of free energy, that dates back to Landau, offers a clear picture of it. Consider an observable ζ(s), function of the system configuration s, that describes the advancement of the nucleation process (or any activated process). As an example, in the Volmer-like theory, s is the volume and surface of the bubble; a candidate for ζ(s) is the radius of the bubble. The free energy profile as a function of its value Z, in our case the grand potential Ω(Z), is related to the probability that ζ(s) = Z through: Ω(Z) = −kBT ln p(Z)

(1)

where kB is the Boltzmann constant and T is the temperature. In other words, the Landau free energy is the probability of a macroscopic state described by ζ(s) = Z measured in kBT units and on a logarithmic scale. The metastable states of the system correspond to the local minima in the free energy profile, i.e., maxima of the probability. The relative stability among metastable states depends on the depth of the minima, which is related via eq 1 to its probability: the thermodynamically stable state corresponds to the absolute minimum of the free energy. Cavitation is observed for pressures below the liquid− vapor coexistence, Pl − Pv ≡ ΔP < 0; in this region the thermodynamically stable state corresponds to vapor. However, organic liquids30 and ultrapure water19,20 can withstand large negative pressures without rupturing, as they are trapped in metastable (but very long-living!) states. The time that the system can stay trapped in the liquid metastable state depends on the free energy barriers ΔΩ† separating it from the vapor state via an Arrhenius-like expression:31 14874


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Figure 1. Illustration of the graphical method for free energy profiles: Geometries of the studied surface defects and bubble nucleation path (left), plot of the curvature versus volume of vapor (center), and free energy profiles constructed via the graphical method for various nucleation numbers Nnu ≡ − LΔP/γlv (right). Three geometries are considered, a square two-dimensional groove (a), a wide three-dimensional conical crevice (b), and a narrow conical crevice (c). The contact angle is assumed to be θY = 110°. Dimensionless quantities are used throughout: υ̃v = Vv/Ld, where d is the dimensionality of the system, j ̃ = JL, and ω̃ = Ω/(γlvLd−1). The reference state for all free energies is assumed to be that where only the liquid phase exists (Wenzel), implying that Δω̃ (0) ≡ ω̃ − ω̃ W = 0. Note that the red curves in the center panels monotonically tend to zero, and, consequently, in the free energy profiles, a maximum and an absolute minimum are always present, even if they do not appear on the scale of right panels. †

Γ = Γ0e−ΔΩ /(kBT )

advancement of nucleation. This is obtained by minimization of the free-energy functional (e.g., the grand potential referring to a system at constant chemical potential μ, temperature T, and control volume Vtot) with respect to the macroscopic system configuration at fixed Vv.32 By repeating this minimization for all Vv between the fully wet Wenzel state33 (Vv = 0) and the pure vapor phase, the set of minimal system configurations as a function of the progress variable Vv is obtained. We interpret this set as the nucleation path, that is, the collection of the most probable configurations encountered along the process of nucleation. In the following, a sharp interface model is assumed for the grand potential, but other formulations, such as the density f unctional theory, may be used in connection with CREaM. With our choice, the specific expression for the grand potential is

(2)

where Γ, the rate of the cavitation process, is the inverse of this time. The exponential dependence of Γ on the free energy barrier, together with the extremely small value of kBT (around 4 × 10−21 J at ambient temperature), makes nucleation extremely sensitive to small changes in free energy barriers. For this reason, the precise determination of the prefactor Γ0 in eq 2 is numerically less important and will be postponed to future work. The main difficulty in nucleation theories is that ΔΩ† has a strong dependence on the thermodynamic conditions, making predictive theories very challenging. In most cases of practical relevance for cavitation, where nucleation is catalyzed by surface defects, a general way to reconstruct free energy barriers is still lacking. In the following, we apply the Continuum Rare Event Method (CREaM), developed according to the theory described in a previous paper,32 to heterogeneous nucleation, trying to fill this gap and showing that an extremely rich phenomenology exists. CREaM is a general approach to find the most probable configuration of a liquid−vapor-solid system at a given volume of vapor Vv, the progress variable of choice to describe the

Ω = −PV l l − PvVv + γsvA sv + γslA sl + γlvA lv

(3)

where P denotes the pressure of the liquid (l) and vapor (v) phases, V is the volume of each phase, γ is the surface energy of an interface between two phases, and A is the surface area of that interface; the subscript s denotes the solid phase. The total volume of the system is constant and given by the sum of the 14875


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volumes occupied by the liquid and vapor phases, Vtot = Vl + Vv = const; similarly, the total area of the solid wall is Atot = Asv + Asl = const. Therefore the grand potential depends on one independent volume and two independent area terms, say Ω(Vv,Alv,Asv). In order to obtain the system configuration that minimizes the grand potential holding the volume of vapor at the fixed value Z, we constrain the grand potential with a Lagrange multiplier λ acting on Vv, I ≡ Ω − λ(Vv − Z). By calculating the first variation of the constrained functional I with respect to the system configuration and setting it to zero, δI = 0, the conditions for stationarity are obtained, namely32 ΔP − λ = Jγlv cos θY = Vv = Z

constructed from these minimal solutions is unique for all ΔP. To better illustrate CREaM, we apply it to identify the nucleation path on a square two-dimensional groove, see Figure 1a (left). The menisci along the nucleation path are circular arcs of different radii intersecting the solid walls with the Young contact angle. The first bubble (in green) nucleates in one of the groove’s corners, and is denoted as asymmetric solution. At higher Vv, the meniscus reaches a symmetric configurationin cyan, symmetricand climbs up the groove with fixed curvature. Once the outer edge of the groove is reached, the triple line is pinned at the groove’s corners (in blue, pinned solution), and the curvature of the meniscus is allowed to vary until the Gibbs’ condition is fulfilled.35 Finally, the triple line is depinned from the sharp corner and the meniscus grows as a circular segment (in red, outside), until the whole available space is filled with vapor. Having identified the nucleation path and taking advantage of eq 4a and of the physical meaning of λ, we now devise a graphical method to reconstruct the free energy profile for bubble nucleation. The first step consists in plotting the curvature of the meniscus against volume of vapor as extracted from the nucleation path. An example of such a plot is reported in Figure 1 (center) for three defects having different geometries, namely, the 2D groove just introduced and narrow/wide crevices (see Figure 1, left). Before presenting the graphical method, let us make all the relevant quantities dimensionless. This allows to describe the different nucleation scenarios based on the sole dimensionless nucleation number, Nnu ≡ − LΔP/γlv. Here L is the characteristic size of the defect defined in Figure 1 (left). The dimensionless version of eq 4a reads

(4a)

γsv − γsl γlv

(4b) (4c)

These conditions are as follows: a modified version of Laplace equation, eq 4a; the usual Young equation for the contact angle, eq 4b; the volume constraint, eq 4c. In eq 4a, J = 1/R1 + 1/R2 is twice the mean curvature of the liquid−vapor interface expressed in terms of the two principal radii of curvature R1 and R2; the mathematical role of λ, the additional term in the Laplace eq 4a, is to enforce a given Vv; it is easily shown that λ = ∂Ω/∂Z along the solutions of eqs 4.32 The standard Laplace equation ΔP = Jγlv defines regular stable and metastable states of a capillary system; once the thermodynamic conditions, determining ΔP and γlv, are given, there is only one possible curvature of the liquid−vapor interface, J = ΔP/γlv. The modified Laplace equation, instead, describes also the states along the nucleation process connecting two metastable states, that is, the nucleation path; along this process, the meniscus curvature is allowed to vary away from the Laplace value J = ΔP/γlv, a fact that was already known from independent atomistic simulations.34 As expected, in the stable and metastable states, corresponding to minima in the free energy profile Ω(Vv), eq 4a coincides with the usual Laplace equation, since there λ = ∂Ω/∂Z = ∂Ω/∂Vv|Vv=Z = 0. The solutions to system 4 are sought for among the surfaces having constant mean curvature, as per eq 4a, with prescribed contact angle, eq 4b. The specific solutions are those enclosing the requested volume, eq 4c. At this step, the solution is not unique; the criterion to choose the most probable one to be used in constructing the nucleation path is given by eq 1 and coincides with the solution of minimal free energy. Since all the surfaces enclose the same Vv, their energetic ordering does not depend on the bulk term of eq 3, and thus the minimal configuration is the same at all ΔP. Once the minimal solution is identified, the Lagrange multiplier λ may be calculated as λ = ΔP − Jγlv. The sequence of minimal configurations parametrized by Z constitutes the nucleation path. It is clear by construction that the nucleation path is independent of ΔP. As a consequence, the difference between grand potential profiles of systems at different ΔPs is ΔΩ(Z) = Ω2(Z) − Ω1(Z) = ΔΔPZ, where ΔΔP = ΔP2 − ΔP1. Summing up, the procedure to obtain the nucleation path consists in two steps: • Compute the possible solutions to system 4 at different advancements of the nucleation process by parametrically varying Z; • Identify the nucleation path by selecting, for each Z, the solution having minimal grand potential; the path

λ ̃ = −(j ̃ + Nnu)

where λ̃ = λL/γlv and j ̃ = JL is the dimensionless curvature. The nucleation number is related to the critical curvature by Nnu = −JcrL where Jcr ≡ ΔP/γlv is the curvature for which λ = 0 and the standard Laplace equation is recovered. Since λ = ∂Ω/∂Vv, minima and maxima of the free energy profile are attained at this critical curvature, where j ̃ = −Nnu, or at the end points of the profile’s subdomains. Extrema may be found graphically ̃ v), with ṽv = from curvature versus volume of vapor graphs j(ṽ Vv/Ld (d = 2,3 dimensionality of the system) as those in Figure ̃ v) graph 1 (center). In particular, the intercepts between the j(ṽ and the j ̃ = −Nnu line identify the (regular) extrema of the free energy profile at a given Nnu. Remembering that λ = ∂Ω/∂Vv, ̃ v) < −Nnu and decreases the free energy increases where j(ṽ elsewhere, thus allowing to sketch the free energy diagram by ̃ v) graph (see Figure 1, right). simple inspection of the j(ṽ Applying the graphical method to the 2D groove as done in Figure 1a, it is seen that by varying Nnu the extrema of the free energy change their location and number. For Nnu slightly larger than zero, the first local minimum corresponds to the Wenzel state while the second minimum is a suspended Cassie state36 with the meniscus protruding into the liquid. The third (absolute) minimum is the pure vapor phase that is attained when the vapor phase occupies all the available space, Vv = Vtot. The free energy profile attains its first maximum in correspondence of the cusp, where the meniscus jumps from an asymmetric to a symmetric configuration.32,34 The second maximum is reached outside the groove when the critical curvature Jcr is attained. For larger values of Nnu, two maxima 14876


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are always present, but the first critical bubble is attained in the asymmetric configuration, in one of the corners of the groove. For even larger values of Nnu (Nnu = 3 in Figure 1a, right), corresponding to very negative ΔP, the second maximum disappears, and the remaining one corresponds to the critical bubble formed in the corner. Similar results can be found for different corrugations (see Figure 1b,c and the following paragraphs). Comparison with Molecular Dynamics. In order to corroborate the continuum theory at the nanoscale where nucleation starts, an independent atomistic technique restrained molecular dynamics34,37 (RMD)is used to reconstruct the free energy profile of nucleation on a system equivalent to the 2D groove. The system consists of a LennardJones (LJ) liquid sitting on a “hydrophobic” LJ wall with a square nanogroove. The essential features underpinning the hydrophobic effect are captured by simple liquids such as LJ38 that, given their simplicity and relatively low computational cost, represent the ideal benchmark for our theory. The single atomistic collective variable m used to describe the emptying of the groove is the atom count inside a box surrounding the groove. m is compared with the continuum parameter describing the advancement of nucleation by setting Vv = (Nbox − m)/ρl, where Nbox is the number of atoms when the box is full (Wenzel state) and ρl is the number density of the bulk liquid. The parameters γlv, ΔP, and θY are measured by separate MD simulations and fed into CREaM to compute the free energy profile. The details of the atomistic methods are discussed in the Supporting Information. The resulting profiles in Figure 2 show a remarkable agreement between RMD and

associated with the critical radius, Rc, being of comparable size with the thickness of the solid−liquid interface, di (see Figure 2). In these conditions, the sharp interface model seems to break down. The same system, simulated at different pressures where the maximum is at much higher Rc, shows the formation of the asymmetric meniscus and the same trend of the free energy profile predicted by CREaM (see Supporting Information). The good agreement between continuum and atomistic results suggests that CREaM is valid down to the nanoscale. Consider also that the present continuum model is valid up to approximately the capillary length lc = [γlv/(ρlg)]1/2, when gravity g starts to deform the liquid−vapor interface (for water in standard conditions lc ∼ 2 mm). These observations suggest that the present theory can be used quantitatively for a range of lengths spanning as much as 6 orders of magnitude. In addition, further extensions to include the effects of gravity, dissolved gas, line tension, surface tension dependence on the radius of curvature (via the Tolman length) are easy to implement in the present framework and can further increase its accuracy and range of validity. Effect of Defect Geometry on Nucleation. We discuss here how the geometry of defects alters the free energy barriers connected with nucleation of vapor cavities, and thus the nucleation rate, as compared to nucleation in the bulk and on flat surfaces. For simple cases, some of the qualitative aspects of nucleation enhancement by surface defects have been already identified by Skripov.18 However, the treatment of general defect geometries require a more detailed analysis as counterintuitive results can be obtained.39 The barriers for nucleation in the bulk liquid and on a flat hydrophobic surface are known from classical nucleation theory.16,18,40 In these cases, nucleation proceeds along simple paths: in 3D, a sequence of spheres of increasing radii for bulk nucleation and spherical caps meeting the surface with the prescribed Young contact angle for nucleation on a flat surface. As a result of these nucleation paths, free energy profiles present a single maximum at the critical radius Rcr = 2γlv/ΔP = 2/Jcr, and thus nucleation is a one-step process leading from the metastable liquid phase to the stable vapor phase. In the presence of surface texturing, instead, nucleation may occur along nontrivial paths, with a significant reduction of the free energy barriers and breaking the process into multiple steps as discussed below. In Figure 3 and Figure 4, we report free energy barriers encountered along the nucleation process on the three defects of Figure 1 (2D square groove, wide and narrow conical crevices, respectively) as computed from the respective free energy profiles. To emphasize the effect of heterogeneity, the barriers are reported as relative values with respect to ∼͠ † = ΔΩ†/ΔΩ†, where ΔΩ† = nucleation in the bulk, Δω b b C(d)πγdlv/|ΔP|d−1 is the free energy barrier for nucleation in the bulk, with d the dimensionality, C(2) = 1, and C(3) = 16/3. On surface defects, nucleation can be a two-step process that involves two free energy barriers to go from the pure liquid phase (Vv = 0) to the pure vapor phase (Vv = Vtot) (see Figure 1, right). To clearly identify all the cases, the barriers are labeled after their transition statethe configuration of the meniscus corresponding to the maximum in the free energy profileas shown in the lower panels of Figure 3 and Figure 4. In particular, the notation Inside, or the subscript in, denotes a transition state happening within the defect, while Outside, or the subscript out, a transition state outside it. Summing up, the

Figure 2. Free energy profiles reconstructed via CREaM (solid line) and restrained molecular dynamics34,37 (RMD, points). The free energy is measured in kBT units. The parameters of RMD simulations used in the continuum model are ΔP = −0.062, T = 0.8, θY = 110°, γlv = 0.55, and L = 10. Lennard-Jones units are employed for all quantities; see the Supporting Information for details. Standard error propagation is used to quantify the uncertainty introduced in the model by errors in the measurement of these parameters (dashed lines). The inset in the main graph shows an enlarged view of the region near the first maximum. On the right, atomistic configurations extracted from the three points around this maximum, showing (Vv increases from top to bottom) the local depletion of atoms with thickness di, the critical bubble with radius Rc, and the symmetric meniscus discussed in the text.

CREaM. We stress that the model is not fitted on RMD data. A qualitative difference, however, can be observed in correspondence of the first maximum shown in the inset in Figure 2. The location of the maximum, nonetheless, is very close to the continuum theory and the height of the barrier is also comparable. Probably the reason of the discrepancy is 14877


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cavitation to occur; the height of each barrier, however, is lower than that needed to nucleate on a flat surface. In other words, the groove acts as a catalyst breaking the nucleation “reaction” into two elementary steps with lower activation energy. In this analogy, the reaction intermediate is the Cassie state. For Nnu < 1.2, the largest barrier is that connected to the Cassie-to-vapor ∼͠ † shown in black in Figure 3) while the transition (Δω out ∼͠ † , in green). Of course, for Wenzel-to-Cassie one is lower (Δω in a groove having zero extension, the free energy barrier coincides with that of a flat surface. A further aspect is that the Wenzel-to-Cassie transition may occur in two different modes, identified by the respective transition states. The first one, active at small nucleation numbers, Nnu < N*nu,1(θY) (N*nu,1 ∼ 1 for the contact angle used, θY = 110°), is associated with the meniscus jumping from an asymmetric to a symmetric configuration (dashed line in Figure 3), in correspondence of the cusp in the free energy diagram. In this range, the free ∼͠ † grows with N . This somehow counterenergy barrier Δω in nu intuitive trend is induced by the matching of the symmetric and asymmetric branches of the free energy profile at increasing ω * , with a resulting growth of the related barrier. When Nnu > Nnu,1 a second nucleation mode takes over, characterized by the formation of the critical bubble in one of the concave corners of the groove. There is a limit nucleation number, N*nu,2 beyond ∼͠ † vanishes, and only the which the Cassie-to-vapor barrier Δω out nucleation-in-the-corner mode remains active. This means that the nucleation becomes again a one-step transition, but with much lower energy barriers than on a flat surface. The second geometry we consider is a three-dimensional conical crevice, a case much studied in literature.25 Crevices may be classified as wide (narrow) if the cone angle β is greater (lower) than 2θY−π (see Figure 1b,c, left). Similarly to the groove, wide crevices show both two-step and one-step nucleation, for the ranges of Figure 4a shaded in gray and white, respectively. This analysis confirms the results by Lu and Peng41 and, through the introduction of the nucleation number in Figure 4, helps to interpret these results in a comprehensive framework. For low nucleation numbers (defined here based on *  −4 sin[θY − (β + π)/ the crevice diameter), 0 < Nnu < Nnu,1 2], a single free energy barrier exists, with the critical bubble ∼͠ † . In this case, the energy formed outside the crevice, Δω out barriers are only slightly lower than in the flat surface case, and, as expected, coincide with it for Nnu = 0. The intermediate Cassie state is possible when the nucleation number falls in the * < Nnu < Nnu,2 *  −4 sin θY, where the pinning region, Nnu,1 curvature j ̃ decreases, as shown in the central panel of Figure 1b. In this range, nucleation is a two-step process with the first ∼͠ † ) significantly lower than the second nucleation barrier (Δω in † ∼ ∼͠ † > Δω ∼͠ † ͠ one (Δω out). A small range of Nnu exists where Δω in out before the Cassie-to-vapor barrier vanishes altogether at Nnu,2 * . * the critical curvature is attained only inside the For Nnu > Nnu,2 crevice, with nucleation becoming again a one-step process with low activation energy. For narrow crevices, yet a different scenario arises as shown in Figure 4b. Here, at negative ΔP, the Wenzel state is unstable, and the pinned Cassie state is the local minimum, as shown by all free energy profiles of Figure 1c. A single maximum exists, attained for bubbles grown outside the crevice; consequently ∼͠ † . The value of this barrier is the only free energy barrier is Δω out practically indistinguishable from that of the wide crevice. For

∼͠ † encountered Figure 3. Relative forward free energy barriers Δω along the nucleation process of vapor cavities from a two-dimensional groove as a function of the nucleation number. Logscale is used for the y axis. All surfaces have a Young contact angle θY = 110° with the liquid. The area shaded in gray between 0 and N*nu,2 identifies the range where nucleation is a two-step process, that is, where two barriers need to be overcome in order to have cavitation from the groove (see center panel). The Inside barrier presents two modes: The first, for 0 < Nnu < N*nu,1, happens at the cusp of the free energy profile (see case Nnu = 0.7 in Figure 1a, right), and is plotted with a dashed green line. In the second mode, for Nnu > N*nu,1, the critical radius is reached when the bubble is in the corner (solid green line). Barriers for nucleation from the bulk liquid (blue) and from a flat surface (red) and are also plotted * a single free-energy barrier is present, for comparison. For Nnu > Nnu,2 and the transition state is inside the groove (bottom panel).

∼͠ † are reported in relative free energy barriers for nucleation Δω Figure 3 for the 2D square groove (green and black lines) as a function of the nucleation number Nnu = −LΔP/γlv; these free energy barriers are compared to those connected with nucleation in the bulk (blue line) and on a flat hydrophobic surface (red line), always in two-dimensions. In Figure 4a, the barriers for nucleation from a wide conical crevice (green and black lines) are compared to 3D nucleation in the bulk (blue line) and on a flat hydrophobic surface (red line). Finally, in Figure 4b, the barrier for nucleation from a narrow conical crevice (black line) is compared to 3D nucleation in the bulk (blue line) and on a flat hydrophobic surface (red line). For all cases, black lines, always present, denote the Outside step of nucleation, while green lines denote the Inside one (see the illustration in the lower panels of Figure 3 and Figure 4). The first geometry we consider is the 2D groove that has been introduced earlier. For 0 < Nnu < N*nu,2 ≡ 2 sin θY, nucleation is a two-step process, involving three (meta)stable states: Wenzel, Cassie, and vapor (shaded area in Figure 2). Two free energy barriers need to be overcome in order for 14878


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∼͠ † encountered along the nucleation process of vapor cavities from crevices. (a) Wide crevice. The Figure 4. Relative forward free energy barriers Δω * < Nnu < Nnu,2 * ) identifies the region where nucleation is a two step process. In the other regions only one barrier is present and the shaded area (Nnu,1 transition state is inside the crevice for Nnu < N*nu,1 and outside the crevice for Nnu > N*nu,2. (b) Narrow crevices. For Nnu < N*nu,2 nucleation is a one * no metastable state exist and the only stable state is vapor. step process with transition state outside the crevice while for Nnu > Nnu,2

Nnu > N*nu,2, the barriers vanish altogether, meaning that the narrow crevice has become an active site for spinodal transition. A further comment is in order for hydrophilic crevices. Even in this case, there are contact angles and nucleation numbers for which a catalytic effect is evident. This observation extends the range of surface chemical properties available for promoting cavitation (see Supporting Information for details). For all the considered cases, at nucleation numbers close to zero, i.e., very small defects, the free energy barriers approach those of a flat surface having the same chemistry. This result highlights that, for nucleation purposes, there exists a cutoff scale, which depends on Nnu, below which the surface details may be neglected and the surface may be considered locally flat. This scale is not purely geometric, as it depends on the thermodynamic conditions via ΔP and on the wetting liquid via γlv. Kinetics of Nucleation of Vapor Cavities. Consider a possible nucleation scenario, with pure water at ambient temperature on a hydrophobic surface (θY = 110°) featuring a periodic pattern of wide conical crevices having period Lc. In these conditions, γlv = 72 mN and kBT = 4.11 × 10−21 J; the system is subject to a constant ΔP. In the following, the probability of reaching the pure water vapor phase starting from the Wenzel state is investigated. The same reasoning may be repeated for any pure liquid/vapor, as, e.g., for organic liquids,30 given the liquid−vapor surface energy and the contact angle. The effect of dissolved gas, which is known to be relevant for water in engineering contexts, is treated in the next section. The total nucleation rate constant for a single step of nucleation is obtained by multiplying the unit rate in eq 2 by the number n of nucleation sites, k = nΓ. In the case of crevices,

the number of defects is given by the surface area A of the system divided by the unit cell area L2c , ncrev = A/L2c . In the case of nucleation on a flat surface, the total area of the system is measured in units of the critical bubble, nflat ∝ A/Rcr2. Nucleation in the bulk can safely be neglected, since the free energy barriers are much higher in this case: in order to obtain the same nucleation rate k as on the flat surface at ΔP = −100 MPa, a volume having the same surface area and a height h > 1040 m would be required. In general, the rate constants k characterizing each nucleation step enter a master equation that describes the evolution of the probability of the metastable states. For a one step reaction A → B, this reads: pȦ (t ) = −kABpA + kBApB

(5a)

pḂ (t ) = kABpA − kBApB

(5b)

where kAB and kBA are the forward and backward rate constants, respectively. Given the exponential dependence of Γand thus kon the free energy barriers, very small differences in ΔΩ† cause one of the two rates to be negligible, and system 5 collapses into a single equation. For instance, if A is the pure liquid and B is the pure vapor phase, even at moderately negative pressures, kAB ≫ kBA since kAB/kBA ∝ exp[−ΔPVtot/ (kBT)], reducing eqs 5 to ṗA = −kABpA. The characteristic time of the system is given by τ = 1/k, where k is now the relevant rate constant (in this example, the forward one, kAB). The probability of the vapor state as a function of time may be obtained solving eqs 5 with the additional condition pB(t) = 1− pA(t). Also in the more complex cases where two nucleation steps are involved (see Figures 1, 3, and 4), it can be 14879


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demonstrated that a single time scale τ matters (see Supporting Information). We are now ready to quantify the catalytic activity of crevice defects, α ≡ τflat/τcrev, comparing the characteristic times on a flat surface and on a pattern of crevices. With this definition, α ∝ Γ0,crev/Γ0,flat(1 − φs)/N2nu exp[(ΔΩ†flat − ΔΩ†crev)/(kBT)], where Γ0 derives from eq 2, φs  1 − πL2/(4L2c ) is the solid fraction of the unit cell containing a single crevice as defined in Figure 5. The absolute free energy barriers are obtained by multiplying the relative ones in Figure 4a by the barrier of the ∼͠ †. corresponding bulk case, ΔΩ† = [π (16/3) (γ3lv/ΔP2)]Δω

cavitation, in particular for water; the phenomenology that arises is very rich and, moreover, it depends on the initial conditions. A complete discussion of this scenario is left for future work; here we sketch the route to introduce the effect of gas into CREaM. We address first the case of a system containing dissolved gas in the absence of contact angle hysteresis and show that the results for the spinodal pressure are in agreement with the Atchley and Prosperetti model for the crevice case.25 Finally, a method is devised to encompass the effect of contact angle hysteresis in the present framework. Suppose there is some gas dissolved in the liquid. Two limit cases arise: (i) the gas diffusion is fast as compared to variations in liquid pressure and (ii) the gas diffusion is so slow that the liquid−gas interface may be considered impermeable to gas.25 The second case seems to be well verified for acoustic cavitation, where the liquid pressure is varied by means of ultrasounds at frequencies up to several megahertz. The CREaM nucleation path for these two limiting cases on surface defects of the kinds considered above can be solved exactly, and these are the cases that we discuss in the following. The intermediate cases between (i) and (ii), in which the characteristic time of gas diffusion is comparable with that of nucleation, can also be solved by CREaM by introducing an extra observable accounting for the chemical composition of the liquid and the gas phase along the nucleation process. The treatment of these more complex cases, however, requires the use of numerical methods, and goes beyond the general illustrative objective of the present work. In the case of fast diffusion, (i), we assume that the system is at chemical equilibrium, and that the partial pressure of gas in the bubble, Pg, is proportional to the concentration of gas dissolved in the liquid, c, through the so-called Henry’s law, Pg = kHc. In these conditions an additional bulk term, −PgVv, is added to the grand potential: Ω = −PlVl − (Pv + Pg)Vv + γsvAsv + γslAsl + γlvAlv, where Vv now is the volume of the cavity filled with vapor and gas. It is apparent from this expression that the theory in case (i) is the same as discussed in the other sections of this work, only with Pv substituted by Pv + Pg. Let us now consider the case in which a fast variation of the liquid pressure is imposed, e.g., by means of ultrasounds. This condition falls in (ii), where the gas is not allowed to diffuse into the liquid. If a bubble of vapor and gas pre-exists in the liquid, then it is found with maximum probability in the metastable states of the system. Therefore cavitation nuclei are assumed here to correspond to the Cassie state. Nuclei with different initial volumes are possible only if contact angle hysteresis is present, as explained below. As in other models,25 the gas is assumed to obey the ideal gas law. The work required to expand isothermally the gas in the nucleus is −Pg,0Vv,0 ln(Vv/ Vv,0), a term that must be added to the grand potential instead of −PgVv of case (i). Pg,0 is the initial partial gas pressure in the nucleus and Vv,0 is the initial volume of the nucleus. Free energy profiles with the extra term arising from the compression/ expansion of the gas nuclei are plotted in Figure 6 for the wide crevice defect. Corresponding profiles for the case without gas are reported for reference. It is seen that the incondensable gas has the effect of lowering the free energy barriers for growing nuclei because the gas pressure helps expanding the initial bubble. Instead, for decreasing nuclei, the Wenzel state disappears in connection with the vertical asymptote at Vv = 0; this effect is due to the fact that the gas liquid−gas interface was assumed to be impermeable to gas and thus it is impossible to absorb the nuclei.

Figure 5. The logarithm of the catalytic activity, ln α, induced by a pattern of wide crevices (compared to the flat surface case) is shown in a loglog plot as a function of the crevice width L for φs = 0.001, γlv = 72 mN, and Γ0,crev/Γ0,flat = 1. The solid blue line depicts ΔP = −1 MPa; the dashed red line ΔP = −10 MPa. The inset illustrates the square pattern of cylindrical holes with diameter L and period Lc.

The logarithm of α is reported in Figure 5 for crevices ranging from 1 nm to 1 μm, φs = 0.001, and for ΔP = −1 MPa and ΔP = −10 MPa. It is worth noting that all the prefactors in α, that is, Γ0,crev/Γ0,flat, 1 − φs, and N−2 nu are slowly depending on L as compared to the exponential term. Figure 5 demonstrates that crevices induce an exceptional boost in the nucleation rate, increasing it up to 350 000 orders of magnitude for ΔP = −1 MPa (the plot shows ln α). It is seen that the catalytic activity increases monotonically with the crevice size L, until a plateau value is reached. Clearly, for unrealistically large L the inverse square dependence on Nnu takes over, and α is expected to decrease for L values beyond the range shown in the figure. In the plateau region, the rate-determining free energy barrier is ∼͠ † , which is in fact constant with respect to N (see Figure Δω in nu 4a, and the Appendix for a detailed discussion on the Nnu dependence of the barriers). The plateau is attained at L ∼ 250 nm and L ∼ 25 nm for ΔP = −1 MPa and ΔP = −10 MPa, respectively. The value of α at the plateau decreases with increasing |ΔP|. From a technological point of view, the plateau identifies the maximum catalytic effect produced by crevices; it also defines an upper threshold beyond which increasing the defect size L does not favor cavitation. The approach described for this specific case can easily be applied to more general defect geometries, provided that the free energy barrier are available (e.g., by using suitable numerical solvers for capillary problems to overcome unpractical analytical solutions). In the next section we also show how to include dissolved gas in the system, a feature often encountered in engineering contexts that considerably lowers the tensile strength of the liquid. Effect of Dissolved Gas. The presence of gas dissolved in the liquid is a case of great importance in engineering 14880


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Figure 7. Sketch of grand potential profiles as a function of ΔP. The arrow indicates the trend with decreasing ΔP (ΔP < 0). Each blue profile is characterized by a free energy barrier ΔΩ† = Ωmax − Ωmin. For the specific case addressed in the sketch, the barrier height decreases with decreasing pressure difference. The red curve denotes the limiting Ω profile associated with spinodal cavitation, ΔΩ† = 0.

Figure 6. Dimensionless free energy profiles ω̃ (υ̃v) for nucleation from a wide conical crevice. Three pressures are plotted, as identified by the different nucleation numbers, Nnu, on the right. The surface has a contact angle θY = 110° with the liquid. In blue dashed lines, the profile relative to a liquid with no dissolved gas. In red solid lines, the profile at the same nucleation number with dissolved gas at Pg,0 = 0.2 γlv/L.

pronounced. The spinodal transition takes place when this state is no longer a minimum, that is, when d2Ω/dV2v → 0+ keeping dΩ/dVv = 0 (see Figure 7). The condition d2Ω/dV2v → 0+ is, indeed, equivalent to eq 6. It is worth stressing that predictions concerning the spinodal transition are confirmed by dedicated experiments27 where pressure pulses deterministically excite the nuclei instability. Indeed, the spinodal transition is a deterministic event, while nucleation is a stochastic, activated event characterized by a kinetics of the kind of eq 2. The present formulation allows one to treat both cases. Contact angle hysteresis may be included somehow empirically in the present framework. This is achieved by replacing the Young contact angle θY in eq 4b by a volume dependent term, θ(Vv). In particular, θ(Vv) = θr for Vv > Vv,0, and θ(Vv) = θa for Vv < Vv,0, where θa and θr are the advancing and receding contact angles, respectively. Clearly, in evaluating the free energy profile, this amounts to replacing the surface energy terms in the grand potential. In Figure 8, we report the case of a system with the same geometry and ΔP as in Figure 6 but with θr = 100° and θa = 120°. The left column of Figure 8 refers to cases where Vv,0 < VC, where VC is the volume of vapor related to the Cassie minimum. The right column refers to cases where Vv,0 > VC. Depending on ΔP and Vv,0, the initial condition may or may not be a metastable state. In particular, in the top and middle panels of the left column of Figure 8 (Nnu = 1.5 and 2, respectively) the initial conditions are metastable; For larger negative ΔP (Nnu = 3.0), the initial condition is unstable and the system relaxes to the Cassie state. On the right column, the initial state is unstable for Nnu = 1.5 and 2.0, while for Nnu = 3.0 the initial state is metastable and prevents an immediate evaporation of the sample (see right column of Figure 8). This simple model for contact angle hysteresis provides two caveats: • Contact angle hysteresis does not mean that metastable nuclei can have arbitrary initial volume; for the values normally encountered on hydrophobic surfaces, θa − θr < 20°, in most cases, hysteresis alone is not sufficient to stabilize nuclei in their initial configuration, see, e.g., Figure 8. • Even if contact angle hysteresis is sufficient to produce a local minimum, the escape from this minimum does not necessarily mean cavitation; the nucleus may simply grow

To complete our analysis, in the following we discuss the initial conditions of nuclei expansion. We imagine a thought experiment divided in three phases. First, a rough surface is put in contact with the liquid. Then, the system is allowed to relax to the local metastable state. Finally, the pressure is suddenly lowered. As already noted by Bankoff,42 the amount of gas that remains trapped inside the surface roughness during the first phase depends on the way the surface comes in contact with the liquid. However, during the second phase, the memory of the initial condition is lost, and the system reaches the local, Cassie or Wenzel, metastable state. The path and growth rate of the nuclei, third phase, depends on a master equation of the kind of eq 5 accounting for the slow drift of the system toward the thermodynamically stable state. Thus, in our thought experiment, the way in which the sample is initially prepared is irrelevant to the mechanism and kinetics of the process. Furthermore, insights in the population of nuclei after the relaxation phase may be obtained from the analysis of the relative probability of the available metastable states. The stability of the nuclei during the third phase (after the pressure is lowered) can be analyzed as follows. In terms of free energy profiles, the spinodal pressure is by definition attained where the free energy barriers vanish, ΔΩ† = 0. This energy condition is equivalent to the mechanical criterion used in the crevice model by Atchley and Prosperetti,25 namely Pg,0Vv,0 ⎞ 2γ ⎞ d ⎛ d ⎛ ⎜P1 + lv ⎟ ⎜Pv + ⎟= ⎝ dR ⎝ Vv ⎠ dR R ⎠

(6)

where R is the radius of the bubble. To show this, we first remark that the Atchley and Prosperetti condition assumes that the Laplace equation (with gas contribution) holds. As already proved in the present article and in a previous work,32 the usual Laplace equation is valid only on regular maxima and minima of the grand potential (i.e., when dΩ/dVv = 0). Of course, the relevant states can only be the minima of Ω. Focusing, for instance, on the metastable Cassie state, suppose that the pressure difference ΔP is decreased, as shown schematically Figure 7; the corresponding minimum (the only one shown in the sketches) in the free energy profile becomes less and less 14881


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Figure 8. Dimensionless free energy profiles ω̃ (υ̃v) for the same system and values of Nnu as in Figure 6 but with contact angle hysteresis, θr = 100° and θa = 120°. The dashed lines denote different choices of initial vapor volumes υ̃v,0. Left column refers to υ̃v,0 lower than the volume of vapor of the Cassie state in the absence of hysteresis.

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to know where cavitation will occur on a surface with engineered patterns, thus realizing “selective” cavitation. The same theory may be employed for estimating the risk of cavitation induced by surface texturing in contexts where cavitation is a detrimental phenomenon. For instance, controlling the effect of geometry is crucial in the design of patterned surfaces for drag-reduction, where one aims at realizing surfaces capable of entrapping air pockets within roughness, but at the same time tries to avoid cavitation damage fostered by the presence of cavitation nuclei.

to the metastable Cassie state that, in turn, will act as a nucleus for cavitation. The activated growth of the nucleus implies cavitation only if no intermediate metastable states are present.

CONCLUSIONS The geometry of surface roughness has a dramatic influence on the nucleation process. Similarly to a catalyst, the presence of surface defects acts in modifying the nucleation path, lowering the free energy barriers separating the metastable liquid state from the vapor phase. We showed that, even on simple geometries, very different nucleation mechanisms are possible, causing a startling increase in nucleation rates that in some cases exceed 105 orders of magnitude with respect to the nucleation rates on flat hydrophobic surfaces. We reported both one-step and two-step nucleation processes with an intermediate suspended Cassie state, and several nucleation modes. These intermediate states correspond to metastable pockets of vapor and/or gas entrapped within surface asperities (in particular, hydrophobic ones) that act as nuclei to accelerate cavitation. A general theory capable of tackling these complex nucleation scenarios on any surfaces was presented, while a systematization of the results was made based on a single dimensionless number, the nucleation number, comparing the intrinsic length scale of surface defects with the critical curvature arising from capillarity. The general theory is developed for vapor nucleation; however, in the last section, the route to include the additional catalytic effect of incondensable gas in the proposed framework is shown. Also, surfaces with contact angle hysteresis are considered. Summing up, the present theory is capable of describing very diverse scenarios, ranging from vapor nucleation to cases where the effect of gas is determining as often encountered in engineering applications. The present results suggest that micro- and nanopatterns may be designed in order to produce controlled cavitation conditions in sonochemistry and biomedical applications. For instance, microfabrication techniques may be exploited to pattern with “smart” defects surfaces or nanoparticles. The geometry of these textures may be designed so as to enhance by orders of magnitude the nucleation rates or require moderate negative pressures in order to produce cavitation. In addition, thanks to the boosting due to corrugations, one is guaranteed

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APPENDIX: DEPENDENCE OF FREE ENERGY BARRIERS ON DIMENSIONLESS NUMBERS Free energy barriers are defined as the free energy differences between two successive minimum and maximum in the free energy profile, ΔΩ† ≡ Ωmax − Ωmin. In Figure 3 and Figure 4, we consider forward barriers (barriers associated to cavitation), i.e., the vapor volume of the minimum is lower than that of the maximum (Vmin < Vmax v v ). In this appendix we explain the dependence of these free energy barriers on Nnu. To achieve this objective we make use of the Buckingham’s π theorem on the dependence of a function on dimensionless numbers. The free energy barriers are, in general, a function of ΔP, of the chemistry of the system (γlv, γsl, and γsv for heterogeneous nucleation), and of the geometry of surface texture through L, ΔΩ†(ΔP,γlv,γsl,γsv,L), see eq 3 and following text. Here, there is no dependence on the variable Vv since the barriers are computed from the free energy profiles at well-defined values of the volume of vapor. Since our system is a mechanical one, there are three fundamental physical units, say length, mass, and time. Thus, by the π theorem, the relative (dimensionless) ∼͠ † depends on 5 − 3 = 2 dimensionless free energy barrier Δω numbers: cos θY = (γsv − γsl)/γlv

(7a)

Nnu = −LΔP/γlv

(7b)

Clearly, for more complex surfaces where more parameters than L are needed to describe the geometry, correspondingly many geometric dimensionless numbers are also defined. For the bulk and flat surface cases, instead, where there is no 14882


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∼͠ † does not depend on L; thus we can define intrinsic length, Δω only one dimensionless number: cos θY = (γsv − γsl)/γlv

(9) Bang, J. H.; Suslick, K. S. Applications of Ultrasound to the Synthesis of Nanostructured Materials. Adv. Mater. 2010, 22, 1039− 1059. (10) Cravotto, G.; Cintas, P. Power ultrasound in organic synthesis: moving cavitational chemistry from academia to innovative and largescale applications. Chem. Soc. Rev. 2006, 35, 180−196. (11) Coleman, A. J.; Saunders, J. E.; Crum, L. A.; Dyson, M. Acoustic cavitation generated by an extracorporeal shockwave lithotripter. Ultrasound Med. Bio. 1987, 13, 69−76. (12) ter Haar, G. Therapeutic Applications of Ultrasound. Prog. Biophys. Mol. Biol. 2007, 93, 111−129. (13) Coussios, C. C.; Roy, R. A. Applications of Acoustics and Cavitation to Noninvasive Therapy and Drug Delivery. Annu. Rev. Fluid. Mech. 2008, 40, 395−420. (14) Unger, E. C.; Matsunaga, T. O.; McCreery, T.; Schumann, P.; Sweitzer, R.; Quigley, R. Therapeutic Applications of Microbubbles. Eur. J. Radiol. 2002, 42, 160−168. (15) Mitragotri, S. Healing Sound: The Use of Ultrasound in Drug Delivery and Other Therapeutic Applications. Nat. Rev. Drug Discovery 2005, 4, 255−260. (16) Volmer, M. Kinetik der Phasenbildung; Theodor Steinkop: Dresden, 1939. (17) Turnbull, D.; Fisher, J. C. Rate of Nucleation in Condensed Systems. J. Chem. Phys. 1949, 17, 71−73. (18) Skripov, V. P. Metastable Liquids; John Wiley and Sons: New York, 1974. (19) Zheng, Q.; Durben, D. J.; Wolf, G. H.; Angell, C. A. Liquids at Large Negative Pressures: Water at the Homogeneous Nucleation Limit. Science 1991, 254, 829−832. (20) El Mekki Azouzi, M.; Ramboz, C.; Lenain, J.-F.; Caupin, F. A Coherent Picture of Water at Extreme Negative Pressure. Nat. Phys. 2013, 9, 38−41. (21) Harvey, E. N.; Barnes, D.; McElroy, W. D.; Whiteley, A.; Pease, D.; Cooper, K. Bubble Formation in Animals. I. Physical Factors. J. Cell. Comp. Physiol. 1944, 24, 1−22. (22) Strasberg, M. Onset of Ultrasonic Cavitation in Tap Water. J. Acoust. Soc. Am. 1959, 31, 163−176. (23) Apfel, R. E. The Role of Impurities in Cavitation-Threshold Determination. J. Acoust. Soc. Am. 1970, 48, 1179−1186. (24) Crum, L. A. Tensile Strength of Water. Nature 1979, 278, 148− 149. (25) Atchley, A. A.; Prosperetti, A. The Crevice Model of Bubble Nucleation. J. Acoust. Soc. Am. 1989, 86, 1065−1084. (26) Chappell, M. A.; Payne, S. J. The Effect of Cavity Geometry on the Nucleation of Bubbles from Cavities. J. Acoust. Soc. Am. 2007, 121, 853. (27) Borkent, B. M.; Gekle, S.; Prosperetti, A.; Lohse, D. Nucleation Threshold and Deactivation Mechanisms of Nanoscopic Cavitation Nuclei. Phys. Fluids 2009, 21, 102003. (28) Aizenberg, J.; Black, A. J.; Whitesides, G. M. Control of Crystal Nucleation by Patterned Self-Assembled Monolayers. Nature 1999, 398, 495−498. (29) Varanasi, K. K.; Hsu, M.; Bhate, N.; Yang, W.; Deng, T. Spatial Control in the Heterogeneous Nucleation of Water. Appl. Phys. Lett. 2009, 95, 094101. (30) Blander, M.; Katz, J. L. Bubble Nucleation in Liquids. AIChE J. 1975, 21, 833−848. (31) Debenedetti, P. G. Metastable Liquids: Concepts and Principles; Princeton University Press: Princeton, NJ, 1996. (32) Giacomello, A.; Chinappi, M.; Meloni, S.; Casciola, C. M. Metastable Wetting on Superhydrophobic Surfaces: Continuum and Atomistic Views of the Cassie−Baxter−Wenzel Transition. Phys. Rev. Lett. 2012, 109, 226102. (33) Wenzel, R. Resistance of Solid Surfaces to Wetting by Water. Ind. Eng. Chem. 1936, 28, 988−994. (34) Giacomello, A.; Meloni, S.; Chinappi, M.; Casciola, C. M. Cassie−Baxter and Wenzel States on a Nanostructured Surface: Phase Diagram, Metastabilities, and Transition Mechanism by Atomistic Free Energy Calculations. Langmuir 2012, 28, 10764−10772.

(8)

Equation 8 shows why the relative free energy barrier of the bulk and flat surface cases is independent of Nnu. Equations 7 show that on textured surfaces, where a characteristic length L ∼͠ † in general depends also on the nucleation number. exists, Δω However, in Figures 3 and 4 we notice that in some ranges of ∼͠ † is Nnu the inside barrier is constant. In particular, Δω in independent of Nnu for the nucleation-in-the-corner mode for the 2D groove and always for the wide crevice. In these cases, the L-dependent terms of the free energy at the transition state are canceled out by corresponding terms at the Wenzel ∼͠ † is the same barrier one would minimum. In other words, Δω in have for bubbles nucleating in corners of infinite extension. Thus, like for nucleation in the bulk and from smooth surfaces, there are ranges of Nnu where the free energy barriers on defects depend only on θY as per eq 8.

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ASSOCIATED CONTENT

* Supporting Information S

Details of the atomistic computations. Nucleation free energy barriers for hydrophilic surfaces. Kinetics for two-step nucleation. This material is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +39 0644585201. Fax: +39 06484854. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC project No. [339446]. S.M. acknowledges the Italian Ministry of Education funding through FIRB grant RBFR10ZUUK. We acknowledge PRACE for awarding us access to resource FERMI based in Italy at Casalecchio di Reno. Further computational resources for RMD were made available by ICHEC (Grant ndphys022a).

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REFERENCES

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(35) Oliver, J.; Huh, C.; Mason, S. Resistance to Spreading of Liquids by Sharp Edges. J. Colloid Interface Sci. 1977, 59, 568−581. (36) Cassie, A. B. D.; Baxter, S. Wettability of Porous Surfaces. Trans. Faraday Soc. 1944, 40, 546−551. (37) Maragliano, L.; Vanden-Eijnden, E. A Temperature Accelerated Method for Sampling Free Energy and Determining Reaction Pathways in Rare Events Simulations. Chem. Phys. Lett. 2006, 426, 168−175. (38) Chandler, D. Interfaces and the Driving Force of Hydrophobic Assembly. Nature 2005, 437, 640−647. (39) In the Supporting Information we show that there are concave defects that can hinder nucleation as compared with the bulk case. In particular, this happens for hydrophilic surfaces (see Figure S4). (40) Kelton, K.; Greer, A. L. Nucleation in Condensed Matter: Applications in Materials and Biology; Elsevier: Oxford/Amsterdam, 2010. (41) Lu, J. F.; Peng., X. F. Dynamical Evolution of Heterogeneous Nucleation on Surfaces with Ideal Cavities. Heat Mass Transfer 2007, 43, 659−667. (42) Bankoff, S. G. Entrapment of Gas in the Spreading of a Liquid over a Rough Surface. AIChE J. 1958, 4, 24−26.

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Geometry as a Catalyst: How Vapor Cavities Nucleate from Defects

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