Heterogeneous Nucleation on Concave Rough Surfaces

Article pubs.acs.org/JPCC

Heterogeneous Nucleation on Concave Rough Surfaces: Thermodynamic Analysis and Implications for Nucleation Design Dongming Yan, Qiang Zeng,* Shilang Xu, Qi Zhang, and Jiyang Wang Institute of Advanced Engineering Structures and Materials, College of Civil Engineering and Architecture, Zhejiang University, Yuhangtang Rd 866, 310058, Hangzhou, People’s Republic of China ABSTRACT: Addressing heterogeneous nucleation process on materials with complex (superficial) structures is an important issue to understand the thermodynamic principles and forward the engineering applications. Based on classic nucleation theory, we developed a thermodynamic model that is comprised of both the geometry and the surface-structure effects of substrate to represent the process of forming a crystal or drop embryo on a rough concave surface. The model recovers many models developed previously. We extracted two featured factors: ratio factor that indicates the ratio of the critical radius of the embryos forming on the rough concave surface to that on the smooth surface, and shape factor that represents the relative free energy barrier of heterogeneous nucleation of the critical size to that of homogeneous nucleation. Anomalies of both the ratio factor and shape factor were observed for heterogeneous nucleation on the rough concave surface. Our study provides an essential route toward design (or manipulation) of heterogeneous nucleation by structuring the surfaces of the substrate in fractal, through which a number of promising applications in a variety of fields can be envisioned.

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INTRODUCTION One of the most important open questions in crystallization is the role played by substrate geometry and surface structure (e.g., topography) with auxiliaries in promoting/depressing crystal nucleation. The features of the controls of heterogeneous nucleation provide clear engineering applications, such as in the development of (super)icephobic surfaces,1 control of ice formation in clouds,2 selection and control of protein crystals,3 and prevention of material damages by ice accretion and adhesion on the surfaces.4 Therefore, understanding the nucleation mechanisms of a phase on a foreign surface and the technology development for further applications are desirable and have been the subject of considerable attention.5−14 Classical nucleation theory (CNT) and exhaust simulations5,6,15−18 predicted a reduced energy barrier in heterogeneous nucleation with/without tight concave features such as scratches, pits, wedges, and grooves. While there is increasing observations evidencing the importance of topography and geometry of substrate in heterogeneous nucleation, there is almost no equivalent support for nucleation on a complex surface structured, for example, in the pattern of self-similarity. A common way to accelerate (or delay) nucleation to form crystals is introducing a foreign surface, of which both the geometry and the topography could be the most important factors that are possible to affect this process. The effect of substrate geometry on nucleation thermodynamics is relatively clear, see for instance the studies by Fletcher,5 Liu et al.,6 Qian, and Ma,7,8 among others. Evidence for the observations that nucleation is more favorable on roughened than on equivalent © XXXX American Chemical Society

smooth surfaces are extensively reported for a range of organic and inorganic substrates.19,20 However, the simple promotion of nucleation with the rougher surface may be challenged by materials with complex structures. Campbell et al.21 conducted a study of the freezing of 50 μm diameter water drops on silicon, glass, and mica substrates and made quantitative comparisons for smooth substrates and those roughened by scratching with three diamond powders of different size distributions. Their findings indicated that surface roughening has no experimentally significant effect on any of the substrates studied. Similar results were reported for crystallization of neo-pentanol and tetrabromomethane on scratched mica surfaces.22 For substrates with much more complex structures, such as porous materials, there may be rather different relationships between the surface structure and nucleation. Recent molecular simulations implied that highly porous materials only act as efficient nucleators for pores of optimal diameters.15,16,18 Simulations (e.g.,23) also suggested that the nucleation promotion ability of the rough surfaces is strongly associated with the surface structures. By advanced molecular simulations, recent investigation by Bi et al.24 revealed that the heterogeneous nucleation of ice on a graphitic surface can be controlled by the coupling of surface crystallinity and surface hydrophilicity. It is noteworthy that the above-mentioned debates are based on the artificial (or simulated) substrates with controlled surface structures. For real materials, of which the Received: February 18, 2016 Revised: April 26, 2016

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DOI: 10.1021/acs.jpcc.6b01693 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C surfaces could be featured in fractal, such as, porous silica,3,25 cement-based porous materials,26,27 and fractured metal10, the effects of the geometry and surface structure on crystallization remained unclear. A fractal surface is structured in the pattern of self-similarity, and this pattern repeats at every scale.28 So nucleation on a such surface should be distinct. Previous studies revealed that the number of nucleation seeds on a fragment with a fractal structure significantly exceeds that on one with flat surfaces as the fractal substrate provides more areas to accommodate the nuclei.3,10 Recently, Zeng and Xu29 established an integral thermodynamic model with parameters that are able to characterize the heterogeneous nucleation process on the fractal surfaces. However, the geometry effect of the substrates, which was noticed by numerous researchers (e.g., Fletcher5), has not been considered in the model. This thus provides scientific incentives to assess the combined effects of geometry and surface structure of substrate on nucleation. In what follows, we, following CNT, first present a section to develop a thermodynamic model of heterogeneous nucleation taking place on a concave rough surface. We showed explicit formulas of size factor defined as the size ratio of the critical embryo on the fractal surface to that on the equivalent smooth surface, and the shape factor following the concept of Fletcher’s shape function. The featured parameters then were analyzed and discussed in-depth. The results may provide new ways to manipulate nucleation process through fabricating appropriate fractal surfaces.

embryo, as the inclination angles of the local contact areas can be different (Figure 1d-1−d-3), we retained a general assumption of the nucleus with the spherical-cap shape, regardless of the geometry and surface properties of the contact substrate (Figure 1a−c). This allows to simplify a practical nucleus−substrate system consisting of complex contact lines and geometries to a well-established system (cf. the nucleation on a concave smooth surface analyzed in ref 8) with a spherical-cap embryo in contact with a rough surface via a mimic interface, which provides a practical way to calculate the volume of the spherical-cap nucleus with complex contact interface. We assumed that the pressure of the parent phase that is generally vapor (or liquid) is uniform and does not change as nucleation takes place, and there is no mass transport in or out of the system. To follow CNT, we adopted the constraints used by Fletcher5 who assumed that the application of macroscopic concepts like surface free energy to very small groups of molecules is acceptable, although the thermodynamic properties may deviate from their macroscopic values, of which the extent is relatively small when thermodynamic inequilibrium (e.g., supercooling or supersaturation) is insignificant. In the present study, we retained the Wenzel-type contact angle interpretation, revealing that the areas of the contact surface are completely covered by the nucleus without empty voids involved. The typical hydrophobic surfaces that may show incomplete contact with the nuclei and thus have empty voids between the phases (described by the Cassie−Baxter type contact angle interpretation) are not considered and are beyond the scope of this study. The regimes of the contacts between the nucleus and substrate were reported to be scale associated. It was suggested the real crystal−substrate interfacial configuration is more likely to be a mix between the Cassie−Baxter state taking place at the larger scales and the Wenzel state at the smaller scales.30 It is thus reasonable to use the Wenzel contact regime in the present study because the nuclei are generally very small (in the magnitude of nanometer). Note that, in the present study, for presentation purpose, the phrase hydrophilicity (or hydrophobicity) represents the acute-angled (or obtuse-angled) contact between an embryo and a surface. For nucleation configured in Figure 1e, the change in the free energy, ΔG, according to CNT, can be expressed by

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THERMODYNAMICS Forming nuclei on rough surfaces is generally very complex, and the process is often represented by tracking a (or an amount of) representative nucleus (nuclei) with definable “macro” properties from solutions, gases, and melts. Following CNT, we focused on the free energy changes ΔG as a representative nucleus is formed and in contact with a substrate surface (Figure 1). Although roughness of the contact surface may make it difficult to identify the contact lines and the spatial geometry of the

ΔG = ΔGv V2 + σ12A12 + (σ23 − σ13)A 23

(1)

where the subscripts 1, 2, and 3 refer to the parent phase, nucleus, and solid substrate, respectively, ΔGv is the free energy change per unit volume of the embryo from phase 1 to phase 2, V2 is the volume of the embryo, σij is the interfacial energy between phases i and j, and Aij is the corresponding interface area. Equation 1 indicates that the free energy change, ΔG, is composed of the (volumetric) energy change of bulk phase transition, ΔGvV2, and the possible energy change as surfaces are created, σ12A12, eliminated, −σ13A23, and replaced by the new interface, σ23A23. This equation always holds under a constant pressure condition. The sphere-cap phase 2 in contact with the concave rough surface consists of the top part, Vt, and the bottom part, Vb (Figure 1e). The volume of the phase 2 thus can be given by

Figure 1. Schematic illustration of a nucleus in contact with a substrate of different geometries and surface properties: (a) planar smooth surface, (b) planar rough surface, (c) concave smooth surface, (d) concave rough surface, and (e) the geometry details of a nucleus in contact with the concave rough surface displayed in (d). With the same Young’s contact angle, θY, three different possible contact regimes may take place for nucleation on the concave rough surface of (d): (d-1) an ellipsoid-cap nucleus on the local surfaces with different inclination angles, (d-2) a sphere-cap (or ellipsoid-cap) nucleus on the local surfaces with an acute inclination angle, and (d-3) that with an obtuse angle.

V2 = Vt + Vb

(2)

If the contact surface is smooth, we can present the volume of the embryo in the bottom part, Vb, the top part, Vt, the area of the created interface between the parent phase 1 and the embryo phase 2, A12, and the area of the eliminated interface between the B


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The Journal of Physical Chemistry C

Table 1. Expressions and Relationships between the Geometric Quantities for a Nucleus Contacted with a Smooth Surface and a Fractal Surface

parent phase 1 and the substrate phase 3, that is replaced by the interface between the embryo phase 2 and the substrate phase 3, A23, in the following expressions, π Vb = R3(2 − 3 cos θ b + cos θ b3) (3a) 3 π Vt = r 3(2 − 3 cos θt + cos θt3) 3

(3b)

A12 = 2πr 2(1 − cos θt)

(3c)

A 23 = 2πR2(1 − cos θ b)

(3d)

σ12 cos θY =

(5)

Replacing eq 3d by eq 4 and substituting eqs 3d and 5 into eq 1, one obtains ⎞ ⎛ A (R) cos θ ⎟ ΔG = ΔGv (Vt + Vb) + σ12⎜A12 − 23 ϕ ⎝ ⎠ =

π ΔGv [R3(2 − 3 cos θ b + cos θ b3) + r 3(2 − 3 cos θt + cos θt3)] 3 + 2πσ12[r 2(1 − cos θt) − R2(1 − cos θ b)cos θ ]

(6)

Equation 6 shows the general free energy change for an embryo in contact with a concave rough surface and remains the same with the expression shown in ref 8. Note that cos θY is constant for a nucleus in contact with a substrate of chemical and physical homogeneity, a differential operation of cos θ (or ϕ) to r thus yields

where R is the radius of the spherical rough void, r is the radius of the nucleus, θ is the apparent contact angle between the nucleus and the substrate, the subscripts “b” and “t” refer to the bottom and top part of the phase 2. The geometry configuration as shown in Figure 1e yields a distance between the highest top and lowest bottom points that is associated with the nucleus size r, the rough void size R and the apparent contact angle, θ, via d = (R2 + r2 + 2Rr cos θ)1/2. The geometry requirements also allow to induce the following relationships:8 cos θt = (r + R cos θ)/d, cos θb = (R + r cos θ)/d, R2(cos θ2b − 1) = r2(cos θ2t − 1), 1 − cos2θb = r2(1 − cos2θ)/d2 and 1 − cos2θt = R2(1− cos2θ)/d2. Those may be independent of the roughness of the surfaces. Equations 1−3 are exactly the same with the expression given in ref 8. For the rough substrate, however, eq 3d is required to be corrected by a roughness ratio, ϕ, that indicates the extra roughness effect of the substrate compared with the smooth one, which gives, A 23(R) = A 23(S)ϕ = 2πR2(1 − cos θ b)ϕ

σ12 cos θ = (σ13 − σ23) ϕ

∂ ln ϕ ∂ cos θ = cos θ ∂r ∂r

(7)

Equation 7 reveals that the change of the apparent contact angle to the nucleation size is proportional to that of the roughness ratio to the nucleation size. Note that eq 7 is meaningless if the apparent contact angle, θ, or the roughness ratio, ϕ, is constant. However, if they are scale-dependent, the characteristics of nucleation on the rough surface maybe different, although the free energy change of the system remains following eq 6. In the present study, we considered a special roughness regime of fractal. A fractal surface shows a repeating pattern (or selfsimilar pattern) that displays at every scale, which is known as evolving symmetry.28 For a real substrate surface, the evolving symmetry of self-similar pattern displays in a limited scale. So for the system considered in the present study, the roughness ratio, ϕ, can be expressed by

(4)

where R and S in the brace represent the rough and smooth conditions, respectively. In eq 4, we further assumed the apparent contact angle of an embryo on an ideal rough surface that is in physical and chemical homogeneity and has a negligible body force, line tension, or contact angle hysteresis between the embryo and the substrate. We should also note the differences between the apparent contact angle, θ, and the Young contact angle, θY, that refers to the intrinsic contact angle between the embryo phase 2 and the substrate phase 3. It generally has C

⎛ 2r sin θt ⎞(D − 2) ϕ=⎜ , 2r sin θt < Lu ⎟ ⎝ L0 ⎠

(8a)

⎛ 2R sin θ b ⎞(D − 2) =⎜ , 2R sin θ b < Lu ⎟ L0 ⎠ ⎝

(8b)


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The Journal of Physical Chemistry C Γ(1 − m2) ∂ ln ϕ = ∂x xΨ

where L0 and Lu are the lower and upper boundaries of the length with fractal measurement of the surface respectively, D is the fractal dimension, a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale in measure. Equation 8 is one of the constitutive equations that feature the fractal surface. We emphasized that eq 8 holds only in the region of L0 < 2R sin θ < Lu, because nucleation embryos are generally very small (at least smaller than the upper bound of the length with fractal measurement). If a large nucleus or droplet is considered, for example, contact angle measurement of fractal surfaces with macro spherical-cap solution drops,31−33 a relationship of ϕ = (Lu/L0)D−2, Lu < 2r sin θt = 2R sin θb34 should be used to replace eq 8. We noticed two variables, r and θt, in eq 8a, and one, θb, in eq 8b. For simplification, we thus used the expression of eq 8b in the following calculations, and we had examined that using eq 8a remains getting the same final formulas. Differentiation operation on eq 8 to the embryo size, r, yields −(D − 2)d(R + r cos θ ) ∂ cos θ b ∂ ln ϕ = ∂r ∂r r 2(1 − cos2 θ )

Similarly, the differential expressions of the subterms of eq 6 can be obtained by aid of the functions present in Table 1. Those give x(1 − m2)3 1 ∂Vb = πr 2 ∂r Λ3Ψ

(14a)

1 ∂Vt =2 πr 2 ∂r − ⎡⎣Ψ(x + m)[2Λ2 + (1 − m2)] + (1 − m2) × [Γm + x(1 − m2)2 ]⎤⎦ /(Λ3Ψ)

(14b)

1 ∂A12 = 2 − ⎡⎣Γm[2(x + m)2 + 1 − m2] + (1 − m2) 2πr ∂r × [2Λ2(x + m) + x(1 − m2)]⎤⎦/(ΛΨ)

(9)

(14d)

eq 14, together with eq 6, yields the differential expression of the free energy change to the embryo radius, given by ∂ΔG = πr 2ΔGv ΠΔG(x , m , D) + 2πrσ12 Πσ (x , m , D) ∂r (15)

(D − 2)R(R + r cos θ)(1 − cos2 θ) ∂ ln ϕ = ∂r (D − 2)r(r + R cos θ)(R + r cos θ)cos θ + d 2r(1 − cos2 θ)

where ΠΔG(x, m, D) and Πσ(x, m, D) are the volume-based coefficient and surface-based coefficient, respectively. They can be expressed by

(10)

This equation shows how the fractal dimension, D, the apparent contact angle, θ, and the embryo size, r, affect the roughness ratio, ϕ. Also, with this relationship, the functions that correlate the geometric quantities to the three variable mentioned above (D, θ, and r) can be present explicitly; see the last column of Table 1. Let m = cos θ and x = r/R, the ratio of the radius of the nucleus to that of the cave, eq 10 can be expressed by

ΠΔG(x , m , D) = 2 −

(x + m)[2Λ2 + (1 − m2)]Ψ + Γ(1 − m2)m Λ3Ψ

(16a) Γm[2(x + m)2 + 1 − m2] ΛΨ (1 − m2)(x + m)[2Λ2 + (1 − m2)] − ΛΨ

Πσ (x , m , D) = 2 −

(D − 2)(1 + xm)(1 − m2) ∂ ln ϕ = ∂x x[(D − 2)(x + m)(1 + xm)m + (1 + x 2 + 2xm)(1 − m2)]

−

(11)

Γ[Λ − (1 + xm)](1 − m2)m x 2 ΛΨ

ΠΔG(x , m , 2) = Πσ (x , m , 2) =2−

(12a)

(x + m)(3 + 4xm + 2x 2 − m2) Λ3 = 2 − 3 cos θt + cos θt3

(12b)

Ψ = Γ(x + m)m + Λ2(1 − m2)

(12c)

(16b)

From eq 15, it can be found that the values of (∂ΔG/∂r) are strongly related to the terms ΠΔG(x, m, D) and Πσ(x, m, D). So one can examine the deviation of eq 15 from the classic expressions by investigating ΠΔG(x, m, D) and Πσ(x, m, D). For a concave smooth surface, D = 2, eq 16 can be simplified to

It is noteworthy that the definition of x, termed as the radius ratio, used in this study is different with that used by Fletcher,5 Qian and Ma,7,8 that is, x = R/r, because of the much more complexity in calculation with the latter definition. We found that some (short) terms repeat in the mathematic expressions. So the terms were screened and represented by symbols to simplify the functions, and were given by

d Λ= = (1 + x 2 + 2xm)1/2 R

(14c)

m(1 − m2)[Γ(Λ − (1 + xm)) + x 2(1 − m2)] 1 ∂A 23(R )cos θ = 2πr ∂r x 2 ΛΨ

In this study, the geometry configuration shown in Figure 1e gives the relationships, cos θb = (R + r cos θ)/d and ∂ cos θb/∂r = [−rR(1− cos2θ) + r2(r + R cos θ)cos θ∂ ln ϕ/∂r]/d3, see Table 1. Using the relations above, one can eliminate the middle term ∂ cos θb/∂r, and finally yields

Γ = (D − 2)(1 + xm)

(13)

(17)

Substitution of eq 17 into eq 15, one obtains ∂ΔG = πr(ΔGv r + 2σ12)(2 − 3 cos θt + cos θt3) ∂r

There may have no physical meanings for the terms of Γ and Ψ, while Λ indicates the ratio of the distance between the top and bottom edges of the embryo to the radius of the concave spherical surface. With the help of the definitions termed to eq 12, one can shorten eq 11 in a rather simple expression, that gives

(18)

This is exactly the same with the formula given by Qian and Ma.8 For a planar smooth surface, D = 2 and x = 0 (or cos θt = cos θ = m), the terms ΠΔG(x, m, D) and Πσ(x, m, D) in eq 16 can be shortened to ΠΔG(0, m , 2) = Πσ (0, m , 2) = 2 − 3m + m3 D

(19)


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Figure 2. Plots of the critical radius factor, H(x, m, D), against the apparent contact angle, θ, as the radius ratio x is set as (a) 10, (b) 1.0, (c) 0.5, (d) 0.1, (e) 0.01, and (f) 0.

the complex relationship between the radius and the roughness ratio. This difficulty had been warned by Fletcher.5 We also tried a rigorous analysis approach recently proposed by Qian and Ma,7 it remained failing to find the explicit function as well. However, it would be convenient to compare the current critical radius of the nucleus in contact with the rough surface, rc, to that with the smooth surface, rc0. It thus gives

And then, eq 15 can be reduced to ∂ΔG = πr(ΔGv r + 2σ12)(2 − 3 cos θ + cos θ 3) ∂r

(20)

It was suggested that eqs 18 and 20 are identical if the apparent contact angle, θ, is assumed to be θt.8 Nevertheless, it should be pointed out that a nonfractal surface condition, that is, D = 2, is required for ΠΔG(x, m, D) = Πσ(x, m, D), so does for eqs 18 and 20. Previous study by the same corresponding author29 has proved that ΠΔG(x, m, D) ≠ Πσ(x, m, D) as nucleation takes place on a planar fractal surface, that is, D ≠ 2 and x = 0. So eq 15 cannot be simplified to an elegant equation, such as, (∂ΔG/∂r) = πr(ΔGvr + 2σ12)f(θ) shown in ref 5. Also, the critical embryo size, rc0 = −2σ12/ΔGv, may not hold for the nucleation on the fractal surface. Again, following CNT, eq 15 was derived to pursue the critical nucleation radius, rc, by letting (∂ΔG/∂r)|r=rc = 0. However, it could be very difficult to obtain an explicit function to show how the critical radius is governed by ΔGv, σ12, θ, and D, because of

rc = rc0H(x , m , D)

(21)

where rc0 is the critical embryo size in some special geometric conditions, for example, the spherical embryos by homogeneous nucleation, the spherical-cap embryos in contact with the planar, concave (or convex) smooth surfaces,5,7,8 and rc0 = −2σ12/ΔGv. H(x, m, D) is a coefficient of critical radius, or termed (critical) radius factor, that reflects the matching extent or derivation extent between rc and rc0 H(x , m , D) = E

Πσ (x , m , D) ΠΔG(x , m , D)

(22) DOI: 10.1021/acs.jpcc.6b01693 J. Phys. Chem. C XXXX, XXX, XXX−XXX


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CRITICAL RADIUS An essential factor that characterizes the process of nucleation on either a smooth or a rough surface is the critical radius of the formed metastable crystal embryos. When an embryo forms homogeneously or heterogeneously in contact with a nonfractal surface, the critical radius, according to CNT, is directly dependent on the surface tension between the parent phase 1 and the embryo phase 2, and the changes of free energy for the phase transition, that is, Rc0 = −2σ12/ΔGv. It could be an intrinsic quantity independent of the roughness and geometry of the nonfractal substrate. Similarly, for the process of nucleation on a fractal surface, one of the most representative characteristic parameters could also be the critical radius. For comparison purposes, we use the critical radius factor, H(x, m, D), presented in eq 22. Figure 2 plots the critical radius factor, H(x, m, D), against the apparent contact angle, θ, as the radius ratio, x, and fractal dimension, D, change. Because it is meaningless as H(x, m, D) < 0 in the sense of physics, we thus only plotted H(x, m, D) that are larger than or equal to zero. First view of Figure 2 reveals the significant effects of the radius ratio on the critical radius factor. If the radius ratio, x, is very large, for example, x = 10, which means the radius of the nucleus is 10× larger than that of the spherical rough pore with a predetermined fractal dimension, two singularities emerge at the acute and obtuse contact angles, respectively; see Figure 2a. Singularities of H(x, m, D) to θ suggest instability of the embryos in contact with the fractal surfaces, where small perturbation of the contact angles can cause significant energy fluctuation that may either facilitate or depress nucleation. The singularities of H(x, m, D) at the obtuse regions of θ disappear as x decreases to 1, where the size of the nucleus is exactly the same with the rough pore, but the values of H(x, m, D) converge at 0 as θ = 180°; see Figure 2b. This means the critical radius of the metastable crystal embryo on the fractal surface is infinitesimal compared with the constant Rc0. When x is lower than 1, all curves of H(x, m, D) converge at 1 under the complete hydrophobic condition, that is, θ = 180°. This may imply that, under the complete hydrophobic contact condition, both the geometry and structure of the substrate have no influence on the metastable nucleation size that is exactly the same with that predicted by previous CNT approaches (e.g., the model by Fletcher5). On the contrary, to the disappearance of the singularities of H(x, m, D) in the obtuse regions of θ, θ ∈ (90, 180], it always shows singularities of H(x, m, D) in the acute regions of θ, θ ∈ [0,90), independent of the radius ratio, x, and fractal dimension, D; see Figure 2a−f. However, the shape of H(x, m, D) around the singularities in the acute regions of θ are significantly different at the different radius ratios and fractal dimensions. If x > 1, H(x, m, D) increases from zero to infinity for all fractal dimensions as θ increases from zero to the singular point, θsing; after that, H(x, m, D) increases from negative infinity to a certain value as θ exceeds θsing. However, when x ∈ (0,1], the curves of H(x, m, D) are rather complex. Consistent increases of H(x, m, D) from zero to infinity with θ increasing are only observed as the radius of the nucleus is not significantly smaller than that of the spherical fractal void, and D is relatively large. Otherwise, H(x, m, D) shows a “Λ”-like shape, that is, it first increases to a certain value and then decreases to negative infinity as θ approaches θsing; see Figure 2b,c. Generally, H(x, m, D) decreases from infinity to 1 when θ increases from θsing to 180°. Two minima of H(x, m, D) appear

It can be easily realized from eqs 17 and 19 that for planar and concave (or convex) smooth surfaces, D = 2, H(x, m, 2) = 1. However, if an embryo is in contact with a fractal surface, D ≠ 2, so H(x, m, 2) ≠ 1. Previous study in ref 29 presented the expression of H as an embryo is nucleated on a planar rough surface with D ≠ 2, x = 0, cos θb = 0, and cos θt = cos θ = m, 3

H(0, m , D) =

2 − 2 Dm + 2(D − 3)m2 + 4m3 − 2

D 5 m 2 3

2 − (D + 1)m + 2(D − 3)m + (6 − D)m − m5

(23)

We then tried to address the free energy required to form a critical nucleus on a fractal concave surface. Substitution of eq 21 into eq 6, one gets, ΔG*|r = rc =

3 16π σ12 f (x , m , D) 3 ΔGv2

(24)

where f(x, m, D) can be cataloged as a shape factor following the concept of shape function by Fletcher5 and expressed by f (x , m , D) = −

−

+

3(1 + xm) ⎛ 1 + xm ⎞3⎤ H3 ⎡ ⎟ ⎥ ⎢2 − +⎜ 3 ⎝ Λ ⎠⎦ Λ 2x ⎣

3(x + m) ⎛ x + m ⎞3⎤ H3 ⎡ ⎟ ⎥ ⎢2 − +⎜ ⎝ Λ ⎠⎦ 2 ⎣ Λ

3H2 ⎡⎜⎛ 1 + xm ⎟⎞ m ⎤ x + m ⎟⎞ ⎜⎛ − 1− ⎥ ⎢⎣⎝1 − ⎠ ⎝ 2 Λ Λ ⎠ x2 ⎦ (25)

A direct and obvious difference between f(x, m, D) and the shape factor in the literature (e.g., refs 5, 7, and 8) is the fractal dimension, D, that features the structure of the contact substrate. Again, for a planar and smooth surface, x = 0 and D = 2, the shape factor can be reduced to f (0, m , 2) =

1 (2 − 3m + m3) 4

(26)

If an embryo was nucleated on a concave smooth surface, x ≠ 0 and D = 2, eq 25 can be rewritten as f (x , m , 2) = −

1 1 ⎛ x + m ⎞⎟3 3mx−2 ⎛⎜ 1 + xm ⎞⎟ − ⎜ − 1− ⎝ ⎠ ⎝ Λ Λ ⎠ 2 2 2

⎛ 1 + xm ⎞ ⎛ 1 + xm ⎞3⎤ x −3 ⎡ ⎟ + ⎜ ⎟ ⎥ ⎢2 − 3⎜ ⎝ Λ ⎠ ⎝ Λ ⎠⎦ 2 ⎣ (27) 8

This is exactly the shape factor obtained by Qian and Ma. When nucleation takes place on a planar substrate with fractal surface, that is, x = 0, D ≠ 2, cos θb = 0, and cos θt = cos θ = m, eq 25 can be simplified as29 f (0, m , D) =

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1 [3H2(0, m , D) − 2H3(0, m , D)](2 − 3m + m3) 4

(28)

The formulas presented above may suggest that the fractal nature of a surface, together with the curvature effect, eq 27, may significantly alter the nucleation process. The analysis given below details how the radius ratio, fractal dimension, and apparent contact angle will affect the nucleation process. F


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Figure 3. Plots of the critical radius factor, H(x, m, D), against the radius ratio, x, as the apparent contact angle, θ, is set as (a-I and a-II) 150°, (b) 110°, and (c) 80°; and against the fractal dimension, D, as the apparent contact angle, θ, is set as the same values: (d) 150°, (e) 110°, and (f) 80°.

for θ ∈ (θsing,180] and x ∈ (0,1); the first one is emerged as θ is

minima and maxima from 1 decrease as x decreases, see the following text for the detailed discussion. When x is decreased to 0, the shape of H(x, m, D) is significantly different from those with x > 0; see Figure 2f. Singularities of H(x, m, D) remain appearing for θ ∈ [0,90), but

just slightly larger than θsing, and the second one as θ is around at 150°; see Figure 2c−e. It can be also found that a maximum appear as θ is around at 110°. The deviation extents of the G


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Figure 4. Changes of the apparent contact angle, θ, of singularity and H(x, m, D) = 0 against the radius ratio, x, for (a) hydrophilic nucleation with x ∈ (0.01,10] and (b) hydrophobic nucleation with x ∈ (1,10].

As singularities can always be observed in H(x, m, D) against θ (Figure 2), we thus aimed to expose the distributions of the singular points and the effects of radius ratio, x, and fractal dimension, D. Meanwhile, for practical physical requirements, again, H(x, m, D) ≥ 0 must be held. Both requirements may help to determine the available domains where nucleation is likely to take place. Figure 4 displays the plots of the apparent contact angle of the singularities and H(x, m, D) = 0 against radius ratio at different fractal dimensions. When the contact is hydrophilic, mostly, the apparent contact angles of the singularities, θsing, are superposed to those of H(x, m, D) = 0, θH=0, when the radius ratio is relatively small, for example, x ≤ 0.2; see Figure 4a. We observed that θsing and θH=0 begin to move away from each other as x is increased to a value, and the points are termed as the intersections of θsing and θH=0. Apparently, both the θsing and θH=0 are almost constant before the intersections. When x is increased to exceed the intersections, however, θsing begins to decrease linearly and slightly, whereas θH=0 to increase nonlinearly and significantly. Increasing the fractal dimensions of the contact surface tends to augment both the θsing and θH=0. Similar observations can be found for the hydrophilic nucleation on plain fractal surfaces reported in ref.,29 but the values of θsing are slightly lower. It can be further found that the intersections between θsing and θH=0 move to a lower value as D increases from 2.1 to 3. For the hydrophobic nucleation on fractal concave surfaces, both the θsing and θH=0 decrease from 180° to certain values as x increases from 1, for example, to 10; see Figure 4b. Superpositions of θsing and θH=0 are only observed for relatively small areas of x > 1. Otherwise, θsing is generally lower than θH=0. Contrary to promoting the effect of fractal dimension on both the θsing and θH=0 for the hydrophilic nucleation, increasing fractal dimension tends to lower the values of θsing and θH=0 for the hydrophobic nucleation. Those suggest that the available domains of the hydrophobic nucleation on fractal concave surfaces are restricted by both the hydrophilicity and fractal dimensions of the substrate surface.

the shape of the curves is completely different. As θ increases from zero to θsing, H(x, m, D) increases from 3 to infinity; and as θ increases further from θsing to 180°, H(x, m, D) increases from negative infinity to 1. We also noted that H(x, m, D) ≈ 1 for θ ∈ (90,180]. Detailed analysis in ref 29 revealed that the maximum deviations that exist at ≈110° are less than 5%. From Figure 2, we noticed the significant effects of the apparent contact angle, θ, on the critical radius factor, H(x, m, D), and some contact angles where H(x, m, D) shows distinct and unique values that may characterize the behavior of H(x, m, D). We thus selected 150, 110, and 80° as the representative apparent contact angles to evaluate the coupled effects of the radius ratio and fractal dimension on the critical radius factor, H(x, m, D). Again, very complex curves of H(x, m, D) against x are shown for θ = 150°; see Figure 3a-I and a-II. Singularities were observed as the radius of the crystal embryo is larger than that of the fractal pore, and the shape of H(x, m, D) and the corresponding singular points change with the fractal dimension. We found that H(x, m, D) tends to merge together when x and D are approaching 2 and 2.12 respectively; see Figure 3a-II. At this apparent contact angle, consistent decreases of H(x, m, D) with the fractal dimension increasing can be observed for all the selected radius ratios, and the decreasing extents are enhanced as x is increased from 0.01 to 1; see Figure 3d. Note that the complex plots of H(x, m, D) against D around the singular points are abandoned as the physical meanings of the data remain vague. When the apparent contact angle was set as 110°, the plots of H(x, m, D) against x and D become much simpler. H(x, m, D) first increases slightly to a maximum value as x is increased from 0 to around 1.5, then decreases rapidly as x is further increased; see Figure 3b. All curves intersect around 3. So, if x < 3, the increasing extents of H(x, m, D) against D are generally promoted; if x > 3, the opposite tendency can be observed; see Figure 3e. When the apparent contact angle is decreased to 80°, monotonous and consistent decreases of H(x, m, D) against x and D take place; see Figure 3c,f. Generally, increasing fractal dimension and radius ratio can increase the deviation extents of H(x, m, D) from 1. It is noteworthy that, for both θ = 110° and 80°, small perturbations of H(x, m, D) to D occur when x is small, for example, x ≤ 0.1. We also noted a characteristic apparent contact angle of 90°, at which H(x, m, D) = 1 whenever x and D vary. This is because ΠΔG(x, m = 0, D) = Πσ(x, m = 0, D) = 2 − x(3 + 2x2) (1 + x2)−3/2 according to eq 16. If the mathematic interpretation is right, this may suggest that both the pore size and surface fractal dimension would not affect the critical size of the formed embryo when the apparent contact angle is a right angle.

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SHAPE FACTOR A distinct feature noted from this development is that the apparent contact angle between the embryo phase 2 and the substrate phase 3 is a function of the radius ratio, x, and the fractal dimension, D, and so does the roughness ratio, ϕ. This feature leads to the very distinct critical embryo size factor, H(x, m, D), that is equal to 1 for nucleation on nonfractal surfaces, and a complex function given by eq 22 for nucleation on fractal surfaces, as analyzed in the previous section. Consequentially, the H


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Figure 5. Plots of the shape factor, f(x, m, D), against the radius ratio, x, at (a) the different apparent contact angles with the set fractal dimensions of 2 and 2.6, (b) 150°, (c) 110°, and (d) 80° as the fractal dimension increases from 2 to 3 with a stepwise increase of 0.1.

Figure 6. Plots of the shape factor, f(x, m, D), against the apparent contact angle, θ, at (a) the different radius ratios with the set fractal dimensions of 2 and 2.6, (b) x = 0, (c) x = 0.1, and (d) x = 0.5 as the fractal dimension increases from 2 to 3 with a stepwise increase of 0.1.

shape function, f(x, m, D), that indicates the deviation of the heterogeneous nucleation process from the homogeneous

nucleation process, may demonstrate its distinctive characteristics. For simplification, we only analyzed the data of f(x, m, D) I


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Figure 7. Deviation of the shape factor against (a) the radius ratio, x, and (b) the apparent contact angle, θ, at the different fractal dimensions.

angle, θ. Figure 6 shows the curves of f(x, m, D) against θ with different radius ratios and fractal dimensions. For comparison, the shape factor for nucleation on flat surfaces with different wettabilities, that is, eq 26, is superimposed for all panels of Figure 6. The plots of f(x, m, D) against θ for the nonfractal surfaces, that is, D = 2, generally conform to the sigmoidal type curve for all the radius ratios. The results indicate that the hydrophilicity of the contact between the nucleus and surface remains a primary factor affecting the heterogeneous nucleation. When the radius ratio is relatively small, for example, x ≤ 0.2 depicted in Figure 6a, the effects of radius ratio are minor. As reported in ref 8, Qian and Ma suggested using the classic shape function of eq 26 to represent the shape factor if a tiny crystal embryo nucleates on a large void. The size effect should be considered as the radius ratio is increased further. However, if the surface of the substrate is fractal, the sigmoidal plots of f(x, m, D) vanish at the regions of small apparent contact angles regardless to the radius ratio. As exemplified in Figure 6a, the regions of the apparent contact angles where f(x, m, D = 2.6) = 0 are about [0, 48)°, which vary very little with the radius ratio. As the apparent contact angle is augmented to exceed 48°, f(x, m, D = 2.6) is shifted from zero to a certain value, and then is increased consistently and monotonously for x ≤ 0.2, which generally follows classic shape factor. The modeled results reveal that the deviation extents are less than 0.1; see Figure 7b. It is relatively clear that fractal dimension affects the available regions of θ. For example, Figure 6c displays the plots of f(x, m, D) against θ for x = 0.1 with different fractal dimensions. It can be seen that the available regions of θ decay progressively from (38,180]° to (53,180]°, which is consistent with the previous results displayed in Figure 4a. Similar results have been also reported for heterogeneous nucleation on a planar fractal surface,29 see Figure 6b. We noted anomalous behaviors of f(x, m, D) for x ≥ 0.5 in the very hydrophobic cases, for example, θ > 130°; see Figure 6a. A sudden drop of f(x, m, D) occurs as θ is exceeding 130° and the size of the nucleus is approaching that of the void with the fractal dimension of 2.6. More severe hydrophobic contact, for example, θ > 170°, leads to repromotion of f(x, m, D). These cause significant deviation of the shape factor from the classic one; see Figure 7b. The anomalies of f(x, m, D) for the hydrophobic nucleation with relatively large embryo size are owning to the obvious changes of H(x, m, D). We then focused on the effect of the fractal dimension on the shape factor. As an exemplification, we set x = 0.5 and plotted f(x, m, D) against θ with different fractal dimensions; see Figure 6d. Due to the size effect, the values of f(x, m, D) are always lower than those of classic form,

in the domains of x < 1, and present the representative results in Figures 5 and 6. Note that the available regions of the shape factor in our model are required to conform to the conditions of H(x, m, D) ≥ 0 and f(x, m, D) ≥ 0. Figure 5 displays the curves of the shape factor, f(x, m, D), against the radius ratio, x, for the selected representative cases. We first exposed the plots of f(x, m, D) against x at different apparent contact angles or different m values following the demonstrations present by Fletcher5 and Qian and Ma.7,8 When nucleation takes place on a complete hydrophobic surface, that is, θ = 180° or m = −1, f(x, m, D) is always equal to 1, independent of the fractal dimension and radius ratio; see Figure 5a. In this case, heterogeneous nucleation reduces to homogeneous nucleation. Typically, if nucleation occurs on a smooth concave surface, that is, D = 2, f(x, m, D) decreases as x increases, which is exactly the same, with the results reported in ref 8. When the contact between the nucleus and the surface changes progressively from the hydrophobic contact to the hydrophilic one, f(x, m, D) drops from 1 substantially. For comparison purpose, we adopted a typical fractal dimension of porous silica materials, that is, D = 2.6,26,27 and plotted f(x, m, D = 2.6) against x on Figure 5a as well. Generally, there has no significant deviation of f(x, m, D) between the nonfractal and fractal surfaces, except for the strong hydrophobic nucleation. As an example, Figure 5b shows the curves of f(x, m, D) against x as m = −0.866 (θ = 150°) and D changes from 2.1 to 3 with a stepwise increment of 0.1. Again, if the radius of the nucleus is significantly lower than that of the concave void, that is, x < 0.2, fractal dimension has minor effect on f(x, m, D). As x is further increased, f(x, m, D) decreases first to a minimum and then increases when D > 2, which distinguishes the nucleation on fractal surfaces from that on nonfractal surfaces. In Figure 5c,d, we also demonstrated the variations of f(x, m, D) with x under different fractal dimensions when θ are equal to 110° and 80°, respectively. Although the effect of fractal dimension on f(x, m, D) is insignificant along the x domain of (0,1), they remain showing some noticeable differences between the apparent contact angles of 110° and 80°. For instance, the observation that increasing fractal dimension lowers f(x, m, D), as displayed in Figure 5b,d, only holds for x < 0.2 in Figure 5c; otherwise, increasing fractal dimension promotes f(x, m, D), of which the extents are, however, negligible. The results can be further supported by the plots of deviation of f(x, m, D) that is defined as f(x, m, D) − f(x = 0, m, D = 2); see Figure 7a. An important characteristic of the shape factor that has been previously explored by numerous researchers (e.g., Fletcher5) is a sigmoidal type plot of f(x, m, D) against the apparent contact J


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behaviors of f(x, m, D) appearing for the very hydrophilic and hydrophobic cases. The elimination of the nucleation energy barrier in the region of small apparent contact angles, as reported in ref 29 and depicted in Figure 6, may be due to the agglomeration of molecules on the fractal surface that is not needed to form a crystal embryo in the defined geometry, such as the spherical-cap nucleus. So it may be impossible to identify the apparent contact angle by the Wenzel contact regime and, consequently, eq 25 fails to describe the relative energy barrier against forming embryos for the very wetting case. However, if the predicted results are correct, it may imply the possibilities of spontaneous crystallization (or condensation) for the very hydrophilic contact case. A significant reduction of shape factor in the very hydrophobic case as the size of the nucleus approaches that of the concave substrate, to the best of our knowledge, has not been reported elsewhere. Under the premise of the model, the results suggested that nucleation is more likely to take place on a hydrophobic rough surface with a match size. Indeed, the match size effect of heterogeneous nucleation has been reported frequently. Based on molecular dynamics simulations, recent studies (e.g., refs 15, 16, 18, and 23) revealed that the nucleation promotion ability of the rough surfaces is strongly associated with the surface structures. A remarkable promotion of nucleation can be observed only when the characteristic length of the surface structure matches well with the size of the crystal. However, if the geometry match between the surface and the crystal is poor, this kind of promotion disappears and the nucleation rate is even smaller than that on the smooth surface. Christenson and co-workers21,22 also argued the preferred nucleation effects by roughing surfaces based on experimental studies. The results showed counteracting tendency of the roughness (or the size of the diamond powders used to scratch the surfaces) on the nucleation density of neopentanol22 and CBr4 and the median freezing temperature of water on the scratched silicon, glass and mica.21 The results, together with the data in the literature, suggested a preferred nucleation length scale of the order of 10 nm.22 Considering that the most critical nucleation size is in this scale, we may conclude that the size match effect predicted in this study is reasonable. Indeed, Salazar-Kuri et al.36 directly observed that submicrometer protein crystals nucleate and grow in pore corners and rough sides of the pore walls, which evidence that the developed model in the present study could be valid in the macro and submacro scales. We should emphasize that this study does not consider the possible effect of chemical nature of a surface, rather than the surface structure. For instance, Lupi et al.37 reported that atomically flat carbon surfaces tend to promote heterogeneous nucleation of ice, while molecularly rough surfaces with the same hydrophobicity do not. It also should be re-emphasized that, as in refs 7, 8, and 29, the characteristics of heterogeneous nucleation on fractal surfaces discussed are limited to where the classic surface physic laws (e.g., the Young and Wenzel equations) will approximately apply. If the embryo is too small, for example, an agglomeration with few molecules on solid substrate, may not hold the classic surface physic laws, while recent molecular simulations by Cabriolu and Li38 suggested that CNT can predict the heterogeneous nucleation behaviors of supercooled ice on carbon surfaces with fairly good accuracy. We should also note the influences of the shape of the crystal embryos and the wetting extend. For a rough surface, the later one may be more important, as the Cassie−Baxter model could be more practical for partially wet states of rough surface. It should be also noted that the

that is, f(0, m,2). Two anomalous regions of the curves emerge. The first region that is consist of the shifts of f(x, m, D), as previous discussed, is due to the singularities of H(x, m, D). The second region that covers the lowering behavior of f(x, m, D) for hydrophobic nucleation, is in (130, 170)°. In this very region, increasing fractal dimension lowers the values of f(x, m, D), which may suggest a higher possibility of nucleation forming on a rougher surface.

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DISCUSSION CNT predicts that the critical size of nuclei that form either homogeneously in bulk or heterogeneously in contact with a solid substrate keeps constant if the relative saturation of the solute (or gas) is controlled, and is independent of the roughness and geometry of the surface. This is because the roughness ratio, ϕ, in eq 5, is constant at any scale of the substrate in CNT. In other words, the apparent contact angle is independent of the scale of the substrate. In our model, the substrate is structured in a pattern of self-similarity, so the contact areas between the nucleus and substrate are associated with the projected length of the nucleus in a power function, that is, eq 8. This thus links the roughness ratio to the apparent contact angle and geometry of the substrate following a complex pattern, see eq 9. While a large amount of data on the size of stable crystals have been reported in the literature (e.g., ref 35), the reports on the size of critical embryos during homogeneous and/or heterogeneous nucleation are very few, because it is difficult to measure the critical size as the nuclei are metastable. Our model predicted the critical size of the nuclei forming on the substrates with fractal structures. It is interesting to note the singularities of the plots of the size factor, H(x, m, D), against the apparent contact angle, θ. The values of H(x, m, D) are resulted from a fraction with the numerator, Πσ(x, m, D), and the denominator, ΠΔG(x, m, D). In the sense of mathematics, according to eq 22, the singularities of H(x, m, D) take place as the denominator is equal to zero, that is, ΠΔG(x, m, D) = 0. In the sense of physics, however, this suggests that the impossibility of forming a embryo with the critical size on a fractal surface. If nucleation takes place on a nonfractal surface, as analyzed in the section of thermodynamic formulations, it always has Πσ(x, m, D = 2) = ΠΔG(x, m, D = 2) ≥ 0. We also noticed that the geometry of the substrate affects the critical nucleation size. Our model also extended previous work by Zeng and Xu,29 who, following the same approach of CNT, developed a thermodynamic model that describes the nucleation behavior of crystals (or droplets) from liquids (or gas) on a planar fractal surface. Their findings showed that significant deviations of the values of the radius factor, H(x = 0, m, D), arise between the fractal surface and the flat surface as the apparent contact angle is acute, while the deviations become negligible as θ is right and/or obtuse. The results are recapitulated in Figure 2f. Generally, nucleation would be favored when a foreign surface is present as the energy barrier required for forming embryos with size larger than and/or equal to the critical value is lowered. Most calculations on heterogeneous nucleation are concerning to the surfaces that have been previous determined by experiments, so have well-described geometries and structures, for instance, the surfaces in the present wedges such as grooves and conical pits.21,22 And the shape factor remained following CNT. However, when the surfaces are structured in the pattern of fractal, the shape factor is different with that of CNT. The size effect of the concave substrate on the shape factor has been exposed in an in-depth study by Qian and Ma.8 Mostly, our results are in accordance with theirs, but we showed anomalous K



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characteristics of both the nucleating phase and the substrate in their crystalline structures may impact nucleation process,16,24 which leads to “anomalous” crystallization behaviors that are not compatible with CNT. As a result, the nucleation of embryos in very tiny size on fractal surfaces maybe beyond the developed thermodynamic formulas based on the classic thermodynamics. We also did not provide any clue to link the surface structure to the kinetics of nucleation taking place, and the relevant issues deserve rigorous study in the future. However, if the fractal structure and the size of a substrate can significantly change the energy barrier of forming a nucleation embryo (droplet condensation) or enhancing the spread of liquid, we would expect a practical approach that fabricates superheterogneous nucleation surface through manipulation of surface texture with fractal structure, such as, providing effective materials and structures to enhance nucleation of proteins that are notoriously difficult to crystallization.36,39,40

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CONCLUSION In the present study, we aimed to expose thermodynamic principles of heterogeneous nucleation on fractal concave surfaces and understand the influences of fractal dimension and size of the substrate on this process. Following CNT, this study consisted of development of a thermodynamic model based on the change of free energy as heterogeneous nucleation takes place on a concave substrate structured with fractal surface, analyzing the featured parameters, that is, the (critical) size factor, H(x, m, D), and shape factor, f(x, m, D), that are extracted from the formulations and distinct from those of classic nucleation on nonfractal surfaces. We found that the size factor shows two singularities at the acute and obtuse apparent contact angles as the size ratio, x, is larger than 1, and the singularities at the obtuse apparent contact angles disappear as x ≤ 1. Increasing the fractal dimensions, D, tends to augment the apparent contact angles of singularity, θsing, whereas the size ratios, x, show negligible effects. The singularities of H(x, m, D) are intrinsically resulted from the mathematic expression of eq 21. The effect of the size ratio, x, on the shape factor, f(x, m, D), basically follows the results reported in ref 8. We, however, reported anomalous behaviors of f(x, m, D) for the very hydrophilic and hydrophobic conditions. Vanishing of f(x, m, D) in the hydrophilic nucleation cases may indicate that nucleation on a fractal surface can take place spontaneously as θ is relatively small, and minima of f(x, m, D) found in the hydrophobic nucleation cases, which have not been reported elsewhere, may reveal a preferred nucleation size consistent with the size match mechanism. Last but not least, the model may pave the way for manipulating surface texture of a substrate with fractal structure to enhance the ability of nucleation with potential applications in many fields.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +86-0571-8820-6763. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.Y. acknowledges the National Natural Science Foundation of China (Nos. 51379186 and 51522905), and Q.Z. acknowledges the National Natural Science Foundation of China (No. 51408536), and Fundamental Research Funds for the Central Universities (No. 513210*172210251). L


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Heterogeneous Nucleation on Concave Rough Surfaces

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