Inclusion of line tension effect in classical nucleation theory for heterogeneous nucleation: A rigorous thermodynamic formulation and some unique conclusions Sanat K. Singha, Prasanta K. Das, and Biswajit Maiti

Citation: J. Chem. Phys. 142, 104706 (2015); doi: 10.1063/1.4914141 View online: http://dx.doi.org/10.1063/1.4914141 View Table of Contents: http://aip.scitation.org/toc/jcp/142/10 Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 142, 104706 (2015)

Inclusion of line tension effect in classical nucleation theory for heterogeneous nucleation: A rigorous thermodynamic formulation and some unique conclusions Sanat K. Singha, Prasanta K. Das,a) and Biswajit Maiti Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

(Received 23 August 2014; accepted 23 February 2015; published online 12 March 2015) A rigorous thermodynamic formulation of the geometric model for heterogeneous nucleation including line tension effect is missing till date due to the associated mathematical hurdles. In this work, we develop a novel thermodynamic formulation based on Classical Nucleation Theory (CNT), which is supposed to illustrate a systematic and a more plausible analysis for the heterogeneous nucleation on a planar surface including the line tension effect. The appreciable range of the critical microscopic contact angle (θ c ), obtained from the generalized Young’s equation and the stability analysis, is θ ∞ < θ c < θ ′ for positive line tension and is θ M < θ c < θ ∞ for negative line tension. θ ∞ is the macroscopic contact angle, θ ′ is the contact angle for which the Helmholtz free energy has the minimum value for the positive line tension, and θ M is the local minima of the nondimensional line tension effect for the negative line tension. The shape factor f , which is basically the dimensionless critical free energy barrier, becomes higher for lower values of θ ∞ and higher values of θ c for positive line tension. The combined effect due to the presence of the triple line and the interfacial areas ( f L + f S ) in shape factor is always within (0, 3.2), resulting f in the range of (0, 1.7) for positive line tension. A formerly presumed appreciable range for θ c (0 < θ c < θ ∞) is found not to be true when the effect of negative line tension is considered for CNT. Estimation based on the property values of some real fluids confirms the relevance of the present analysis. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914141]

I. INTRODUCTION

Nucleation can be defined as the process of formation of the new stable daughter phase from a metastable mother phase. This phenomenon is ubiquitous in nature and is important in many branches of science and technology. Generally, nucleation is associated with the first-order phase change phenomena such as condensation of a vapor, boiling of a liquid, melting of a crystal, solidification of a liquid, and formation of metallic glass from an alloy melt etc.1 The thermodynamic driving force for nucleation is the difference between the chemical potentials of the metastable phase and the newly formed stable phase. For occurrence of nucleation and subsequent growth of the nucleated phase, a non-zero positive value of this chemical potential difference is essential. The system must overcome the free energy barrier defined by the interfacial energy to form a stable nucleus. If the formation of the new stable phase is solely from within metastable bulk phase, then the nucleation is defined as homogeneous nucleation. On the other hand, in heterogeneous nucleation, formation of a nucleus occurs in the interface between the metastable phase and an extrinsic nucleating agent. The nucleating agent may be a solid or a liquid surface such as dust particles, aerosol, biological macromolecules, or any type of impurities other than the metastable phase. Since these nucleating agents are present extensively in most of a)Electronic mail: [email protected]

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the natural and industrial processes, nucleation is in general heterogeneous in nature. Therefore, the study of heterogeneous nucleation is important for diverse environmental2 phenomena, industrial3 systems, and biomedical4 applications. Despite the dispute pertaining to the various associated assumptions, Classical Nucleation Theory (CNT) is one of the most versatile and widely adopted theories for explaining the heterogeneous nucleation on flat planar surfaces5–7 as well as curved substrates.8–12 Most of the analyses based on CNT ignore line tension. However, line tension cannot be ignored in the incipient stage of the heterogeneous nucleation, i.e., when the nucleus is atomistically small. The line tension arises in the triple boundary line which is the interfacial line where all the three phases meet. From the Gibbsian thermodynamical path, if line tension is considered as a one-dimensional counterpart of the surface tension, then the origin of the excess free energy at the triple line is primarily due to the effect of intermolecular interactions among the molecules situated along the line. The concept of contact line tension or peripheral tension was introduced by Gibbs13 over a century ago, but still a questionable topic considering its magnitude and sign. The theoretical estimated absolute values of line tension are within the range of 10−12–10−10 N, especially experimental values as big as 10−6 N or even higher were reported. However, this has been debated by many researchers in the equilibrium wetting phenomenon.14 Widom15 has described theoretically the effect of line tension on the microscopic or apparent contact angle for a small sessile drop resting on an ideal solid planar

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surface at equilibrium. Marmur16 has considered an approximate thermodynamic theory to determine the value of line tension and microscopic contact angle. Analyzing based on generalized Young’s equation and effective interface potential approaches, Pompe and Herminghaus17 have determined experimentally the contact line tension based on liquid surface topography using scanning force microscope. Wang et al.18 have discussed the value and sign of peripheral tension with the help of microscopic interferometry and their analyses are based on generalized Young’s equation approach and interface displacement modeling. Considering surface heterogeneities and van der Waals interactions, Checco et al.19 have shown the effect of contact line curvature of the liquid droplets on microscopic contact angle using true noncontact atomic force microscopy (AFM) and have also discussed with the help of two approaches, viz. interface displacement modeling and effective interface potential. Linear tension associated with butane saturated water droplet and nitrogen bubble at Si (1 0 0) has been determined by Kameda et al.20 using tapping mode AFM. Based on the positioning of the Gibbs dividing interfaces, Schimmele et al.21 have analyzed thoroughly the effect of the line tension, the Tolman length, and the stiffness coefficients of the triple contact line for a liquid lens and a droplet on equilibrium wetting phenomena. Ruckenstein and Berim22 reviewed the wetting phenomena based on two microscopic approaches, viz. minimization of potential energy and classical density functional theory. As line tension contributes to the free energy barrier, it may play a significant role during heterogeneous nucleation. Heterogeneous nucleation on planar surfaces23–27 as well as spherical substrates28–30 including line tension effect has been examined from time to time. Gretz23,24 was the first to understand the influence of line tension in heterogeneous nucleation. Evans and Lane25 investigated the line tension effect in the theory of heterogeneous ice nucleation. A thorough discussion was made on heterogeneous nucleation including both positive and negative line tension effect by Navascues and Tarazona.26 Considering triple boundary line, Greer27 compared both vapor to liquid and liquid to crystal nucleation on a planar substrate with experimental results. Lazaridis28 studied the effect of surface diffusion and negative line tension for seed mediated nucleation on convex spherical substrate. Hienola29,30 discussed the effect of negative line tension on nucleation of various unary and binary systems and also compared their analysis with experimental results. Due to its generic thermodynamical formulation, the primitive model developed by Gretz23,24 or the modified model by Navascues and Tarazona26 has been widely used in the study of heterogeneous nucleation including the line tension effect. Nevertheless, there are controversies regarding the assumptions of spherical-cap shaped embryo, unfeasible negative critical free energy barrier for nucleation, etc. Essentially, the justification behind the extensive use of this model is that the model assists to explain the heterogeneous nucleation phenomena appearing in different natural processes and industrial systems. However, any model based on a systematic and rigorous thermodynamical critical-value analysis is still absent because of the mathematical complicacies associated with the theoret-

J. Chem. Phys. 142, 104706 (2015)

ical derivation. Essentially, Gretz23,24 or Navascues and Tarazona26 deduced their thermodynamic relation indirectly and did not consider two crucial sufficient conditions to describe the nucleation free energy barrier as will be shown afterwards. This not only has left the model radically inadequate but also has reasonably shrouded its substantial characteristics, which could have helped the improved understanding of heterogeneous nucleation in the presence of line tension. In the present work, using the paradigm of CNT, a geometric model for heterogeneous nucleation on flat planar substrates with the inclusion of line tension effect has been attempted. We present a unique analytical treatment for a rigorous thermodynamic derivation of the critical work of formation in heterogeneous nucleation. Starting from the mechanical equilibrium of the three phases, the critical radius of nucleation and the work needed to achieve it have been derived. Finally, from the stability consideration, the range of critical contact angle for nucleation has been identified over the entire possible range of contact angles for both positive and negative line tensions. The analysis yields a number of new attributes of heterogeneous nucleation on flat surface in the presence of line tension effect and their significance is explained based on macroscopic and microscopic contact angle. The significance of the theory developed has been emphasized using the property data for some real fluids as well as the results of one nucleation experiment.

II. THERMODYNAMICAL FORMULATION A. Geometrical parameters

Consider the formation of a spherical-cap shaped embryo of radius r on a planar surface as illustrated in Fig. 1. A cluster ( β) is formed at the interface between the metastable phase (α) and the substrate (N). O is the spherical center of β. S is the junction where the three phases α, β, and N meet each other. θ is the contact angle between β and N.

FIG. 1. Heterogeneous nucleation of a liquid drop from a metastable phase α on a substrate N : (a) A more stable phase β forms a spherical cap of radius r , where the radius of the triple boundary line is r ′. (b) At the junction S of a planar α–β–N interface, the contact angle θ is determined by the generalized Young’s equation (Eq. (9) or Eq. (10)).

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With the help of above notations, the following geometric quantities and relationships can be obtained trivially. The length of the triple line can be expressed as L = 2πr sin θ.

(1)

The interfacial area between α and β(Aα β ), the interfacial area between β and N(Aβ N ), and the volume (V ) of the spherical-cap β can be obtained from the geometry, Aα β = 2πr 2 (1 − cos θ) ,

(2)

Aβ N = πr sin θ,

(3)

2

2

and V=


π 3 r 2 − 3 cos θ + cos3θ . 3

(4)

B. Minimization of Helmholtz free energy

Helmholtz free energy (F) for the formation of an embryo from the metastable phase can be defined as
F = σ Aα β + σ β N − σα N Aβ N + γL, (5) where γ is the line tension acting tangentially along the triple line and σ, σα N , and σ β N , are curvature-independent and supersaturation-independent interfacial tension between embryo–metastable phase, embryo–nucleating agent, and metastable phase–nucleating agent, respectively. Now, from Young’s equation, the macroscopic contact angle (θ ∞) can be found as σα N − σ β N . (6) σ Substituting the different interfacial areas, triple line length, and equilibrium contact angle from Young’s equation, F can be modified as
 F = σ 2πr 2 (1 − cos θ) − cos θ ∞ × πr 2sin2θ cos θ ∞ =

+ (γ × 2πr sin θ) .

F is the standard Helmholtz free energy of formation of a condensed phase, and it is equivalent to the mechanical work done by forces due to line tension and surface tension. Moreover, the final result obtained in this sub-section is the generalized Young’s equation, i.e., Eq. (9) or (10). This equation can also be rewritten as σ β N − σα N + σ cos θ + (γ/r ′) = 0, which represents equilibrium of the various associated interfacial and peripheral forces for a sessile condensed phase at equilibrium18,27 as depicted in Fig. 1(b). Interestingly, using γ as given in Eq. (9), one may get λ = 2σ r = ∆p. The Lagrange multiplier is therefore the difference of pressure (∆p) between the newly formed phase and the metastable state. It may further be shown that λ is also equal to G, the difference of Gibbs free energy per unit volume between the nucleating and the metastable phase. Intuitively, the line tension appears to be positive in the formulation. However, no thermodynamic principle is violated, as can be seen from Eq. (9) or (10), if a negative line tension is considered in the theoretical formulation. The line tension, particularly its sign, has a significant effect on the microscopic contact angle of a liquid drop on a solid substrate. As mentioned earlier, some intermolecular interactions among the molecules are always present in the vicinity of the triple line. Due to these interactions, the molecules along the contact line will tend to migrate towards a more stable state, so that the free energy of the system becomes the minimum. Moreover, the shape of the liquid surface in the neighborhood of the triple boundary not only changes with the magnitude of the line tension but also with its sign. The microscopic contact angle becomes different from the macroscopic one due to these intermolecular interactions.29 These changes in the macroscopic contact angle are illustrated in Fig. 2 for two cases of heterogeneous nucleation of a spherical-cap liquid drop on a planar substrate with (a) positive line tension and (b) negative line tension. The microscopic contact angle becomes greater than the macroscopic

(7)

Under the constraint of a fixed volume, F is a function of r and θ. F can be minimized using the Lagrange multiplier technique. Introducing the Lagrange multiplier λ, one may write F ′ = F − λV , where  F ′ = σ 2πr 2 (1 − cos θ) − cos θ ∞ × πr 2sin2θ
π + γ × 2πr sin θ − λ × r 3 2 − 3 cos θ + cos3θ . (8) 3 Application of Lagrange multiplier technique Appendix A gives γ = sin θ (cos θ ∞ − cos θ) (9) σr or δ = sin θ (cos θ ∞ − cos θ) . (10) r The above modified equation is the so called generalized Young’s equation for the heterogeneous nucleation considering the effect of line tension, where δ = γ/σ. In this context of energy minimization, the thermodynamic free energy can be defined as the amount of work that a thermodynamic system can perform. In the present case,

FIG. 2. Modifications of the contact angle for a liquid drop on a flat solid surface: (a) positive line tension reduces the perimeter of the contact boundary line which in turn leads to θ > θ ∞; (b) negative line tension enlarges the perimeter of the contact boundary line and leads to θ < θ ∞ and may also form a precursor film.

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one for positive line tension resulting in a reduction in the length of the three-phase boundary line. In this connection, a very interesting result has been demonstrated by Widom15 for a small sessile drop. With positive line tension, in such case, the contact angle may reduce to zero; the droplet is then completely detached from the substrate and remains in contact only with the vapor. On the other hand, the microscopic contact angle will be smaller than the macroscopic one for negative line tension, following an enlargement of the contact line length.29 Moreover, a precursor film may form due to this spreading for negative line tension as shown in Fig. 2(b). Definitely, the variation in the contact angle due to the line tension has a considerable effect on the classical theory of heterogeneous nucleation. Besides, it will be subsequently shown that the microscopic contact angle must be always within an appropriate limit for a possible nucleation. C. Critical parameters in heterogeneous nucleation

From the thermodynamics of phase transition, it can be shown that the change in Gibbs free energy associated with the formation of an embryo from the metastable mother phase is E = F − (G × V ) = F − F ′′,

(11)

where 
F = σ 2πr 2 (1 − cos θ) − cos θ ∞ × πr 2sin2θ + (γ × 2πr sin θ)

(12)

and F ′′ = G ·


π 3 r 2 − 3 cos θ + cos3θ . 3

(13)

The determination of the critical free energy barrier (Ec ) demands dE/dr | r =r c = 0. However, the ab initio mathematical treatment is not straightforward and involves rigorous derivations. This mathematical difficulty compelled the researchers to use any indirect approach to determine the critical free energy barrier. For a seed mediated nucleation over a convex surface, Fletcher9 retained the Gibbs-Thompson relationship for critical radius of homogeneous nucleation, i.e., r c = 2σ/G, even in the absence of the line tension effect. Further, this value of critical radius is substituted in the expression of E to obtain the value of Ec , the critical free energy barrier for nucleation. Accounting for the line tension effect, Navascues and Tarazona26 used the above mentioned indirect method to find the free energy barrier for heterogeneous nucleation on a flat surface. However, their formulation raises two potential queries for the justification of the approach, as critically questioned by Qian and Ma.11 First of all, dE/dr | r =r c = 0 will be a quadratic equation (as it is derived from the cubic Eq. (11)). It is not guaranteed that r c = 2σ/G is the only root of the cubic Eq. (11). Besides, for Ec to be the critical free energy barrier to nucleation, either d 2 E/dr 2 < 0 at r = r c must be satisfied or additional one or more conditions for stability of the system need to be established. In this work, we attempt to answer these questions by implementing a rigorous and systematic thermodynamic critical-value formulation and its implications.

For mechanical equilibrium of heterogeneous nucleation (Eq. (9)), we have Appendix B sin θ (cos θ ∞ − cos θ) dθ =−
. dr r cos θ ∞ cos θ − cos2θ + sin2θ

(14)

As the work for forming an embryo E = E(r, θ), one gets ( ) dE ∂E ∂E dθ ∂F ∂F ′′ = + · = − dr ∂r ∂θ dr ∂r ∂r ( ) ′′ ∂F ∂F dθ · − . (15) + dr ∂θ ∂θ Substituting the derivatives of F and F ′′ in Eq. (15), one finally gets Appendix C 

dE 2 = 2πσr − πGr 2 − 3 cos θ + cos3θ dr ) ( dθ · rsin3θ . (16) + dr Using Eq. (14), for dθ/dr, 
 dE 3 = πr (2σ − Gr) ×   2 − 3 cos θ + cos θ dr    (cos θ ∞ − cos θ) sin4θ (17) −
 . 2  2 cos θ ∞ cos θ − cos θ + sin θ  For heterogeneous nucleation (dE/dr = 0), either or both of the last two factors in the RHS of Eq. (17) need to be zero as πr is non-zero. For 2σ/G to be the sole root of the equation, the last factor has a non-zero value. The stability analysis in Sec. II D not only establishes that the mentioned term is greater than zero but also helps to deduce the appropriate range of θ c for nucleation. For the critical condition given by dE/dr | r =r c = 0, we can write 2σ . (18) G For heterogeneous nucleation on planar substrate, the microscopic and macroscopic contact angles are the same (θ = θ ∞) in the absence of line tension effect. So, for this limiting condition, dE/dr can be rewritten as rc =


dE = πr (2σ − Gr) × 2 − 3 cos θ ∞ + cos3θ ∞ . (19) dr Substituting r c from Eq. (18) in the above equation, the critical free energy barrier (Ec ) for nucleus formation can be obtained as 4πσ 3  Ec = 2 (1 − cos θ c ) − cos θ ∞ sin2θ c G2
4πσγ 8πσ 3 + sin θ c − 2 − 3 cos θ c + cos3θ c . (20) G 3G2 Also substituting the value of critical radius from Eq. (18) in nucleus equilibrium equation, i.e., Eq. (9) or (10), we have γG δ = = y = sin θ c (cos θ ∞ − cos θ c ) . 2 r 2σ c

(21)

Eliminating the terms containing macroscopic contact angle (θ ∞) from the expression of critical free energy barrier

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expression in (Eq. (20)) and using the above equation, we obtain ( ) 3γG sin θ c 16πσ 3 1 3 × 2 − 3 cos θ c + cos θ c + . Ec = 4 3G2 2σ 2 (22) The above equation can be further modified with the help of three dimensionless parameters, namely, m = cos θ c , m∞ γG δ = cos θ ∞, and y = 2σ 2 = r c . Substituting all the mentioned parameters in the expression for free energy barrier, finally we have Ec = Echomo · f (m, y),

point. As discussed in Sec. II C, there is a requirement of stability analysis in the theoretical formulation, which in turn assists to derive the appreciable range of nucleation contact angles. Besides, the stability analysis will not only explore some substantial features but also completes the analytical derivation for heterogeneous nucleation associated with the effect of line tension. Whether E yields a local maximum or not at r = r c = 2σ/G can be verified by checking the sign of d 2 E/dr 2, 
 d 2E 2 − 3 cos θ + cos3θ = 2π (σ − Gr) ×  2  dr 

(23)

  (cos θ ∞ − cos θ) sin4θ 
2 cos θ ∞ cos θ − cos2θ + sin θ   
d   2 − 3 cos θ + cos3θ + πr (2σ − Gr) ×  dr    (cos θ ∞ − cos θ) sin4θ . − (28)
2  2 cos θ ∞ cos θ − cos θ + sin θ 

−

where Echomo =

16πσ 3 3G2

(24)

and √ ) 1 ( × 2 − 3m + m3 + 3 y 1 − m2 . (25) 4 The above mentioned bivariate function can also be written as a function of macroscopic contact angle (θ ∞) and critical microscopic contact angle (θ c ) with the help of generalized Young’s equation, i.e., Eq. (9) or (10), as 1 2 − 3 cos θ c + cos3θ c f (θ c , θ ∞) = 4 + 3sin2θ c (cos θ ∞ − cos θ c ) . (26) f (m, y) =

Substituting r = r c = 2σ/G in Eq. (28), one gets 
 d 2 E 3 = −2πσ ×  2 − 3 cos θ + cos θ 2  dr r =r c  −

Alternatively,




1 f (m, m∞) = × 2 + 3m∞ − 6m − 3m∞m2 + 4m3 . 4 (27) It is needless to say that the function f is of paramount importance as it signifies the critical free energy of nucleation in the presence of line tension effect. It bears the information regarding the shape of the nucleated phase and will be referred as the shape factor in rest of this work. D. Appreciable range of critical microscopic contact angle

Widom15 has described analytically the effect of line tension on the microscopic or apparent contact angle for a small sessile drop resting on a solid planar surface at equilibrium, i.e., equilibrium wetting phenomena. However, a metastable vapor-liquid system is taken into consideration in the present work. Therefore, metastability of the system plays a vital role in this context rather than in the case of equilibrium state. Free energy barrier is one of the most important parameter in the study of nucleation. In this article, a rigorous theoretical analysis of heterogeneous nucleation based on CNT not only considers the effect of line tension but also describes the variation of free energy barrier for nucleation with respect to microscopic and macroscopic contact angle. Moreover, our study includes solution of critical value problem and stability analysis. To ascertain that r = 2σ/G gives the critical radius at nucleation, one needs to check the maximality of the stationary

  (cos θ ∞ − cos θ) sin4θ
 < 0, 2  2 cos θ ∞ cos θ − cos θ + sin θ 

(29)

sin2θ c (2 + cos θ c ) − (cos θ ∞ − cos θ c ) > 0,  2 sin θ c + cos θ c (cos θ ∞ − cos θ c )

(30)

sin3θ c (2 + cos θ c ) − y > 0. sin3θ c + y · cos θ c

(31)

or

The appreciable range of critical microscopic contact angles for positive line tension as well as negative line tension can be found by solving inequality (30) or (31). The inequalities are satisfied if both the numerator (N) and denominator (D) have the same sign. Fig. 3(a) illustrates the values of N and D with respect to θ c for θ ∞ = 45◦. The profiles of N(θ c , θ ∞) and D(θ c , θ ∞) are represented a specific value of θ ∞. As the trends of these profiles are similar for the entire range of θ ∞, the resulting outcomes from these profiles will be shown subsequently in a tabular form for a certain range of θ ∞. Fig. 3(b) represents the variation of y with θ c for θ ∞ = 45◦. It may be noted that mathematically θ c can be calculated for the entire range of macroscopic contact angle, θ ∞. However, nucleation is plausible only within a limited range of θ c . Based on the preceding formulation on the mechanical equilibrium and the subsequent analysis of the stability, the domain of θ c for the present physical problem can be divided into 5 regions, viz. P, Q, R, S, and T, as shown in Figs. 3(a) and 3(b). At this point, it would be instructive to discuss the variation of N and D in different subdomains and also to derive the possible range of nucleation contact angle. As illustrated in Fig. 3(a), the numerator of Eq. (30), i.e., sin2θ c (2 + cos θ c ) − (cos θ ∞ − cos θ c ) = 0 has only one

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FIG. 3. Variation of (a) numerator (N ) and denominator (D) of LHS of inequality (31) and (b) dimensionless line tension y with respect to θ c for θ ∞ = 45◦. From the condition of mechanical equilibrium and stability analysis, the various resulting subdomains are as follows: P—unstable for −ve γ, Q—stable for −ve γ, R—stable for +ve γ, S—unstable for +ve γ, T—physically unstable for +ve γ.

solution in 0◦ < θ c ≤ 180◦ irrespective of θ ∞. The solution denoted as θ ′ is represented by the joining point of the two regions R and S at the abscissa. So, θ ′ must be the upper limit of possible nucleation contact angle for positive line tension, as it has already been shown in Sec. II B that microscopic contact angle must be greater than θ ∞ for positive line tension. One of the appreciable ranges of θ c is θ ∞ < θ c < θ ′ (section R) for a positive line tension. Moreover, it can also be seen from Fig. 3 that both N and D are positive in θ ∞ < θ c < θ ′ for a positive line tension, which further satisfies the stability condition mentioned earlier. Interestingly, one can also get N = 0 from dF/dθ| r =r c = 0 for positive line tension. So, θ ′ is essentially the microscopic critical contact angle for which the

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Helmholtz free energy F has its local minima for positive line tension. It can also be seen from Fig. 3(a) that D = 0 or sin2θ c + cos θ c (cos θ ∞ − cos θ c ) = 0 has two solutions (locations) in 0◦ ≤ θ c ≤ 180◦ irrespective of θ ∞. Eventually, it can also be shown that D = 0 signifies d (δ/r) /dθ = 0. Let us denote the solution of D = 0 is θ M which has two values depending on the sign of the line tension. Clearly, the lower (joining point of the two sections P and Q at the abscissa) and higher (meeting point of the two regions S and T at the abscissa) values of θ M correspond to the negative line tension effect and the positive line tension effect, respectively. θ M < θ ∞ for negative line tension and θ M > θ ′ for positive line tension, as depicted in Figs. 3(a) and 3(b) for θ ∞ = 45◦. Therefore, the possible range of θ c is θ M < θ c < θ ∞ (subdomain Q) for negative line tension. Moreover, both N and D are positive in θ M < θ c < θ ∞ for negative line tension as depicted in Fig. 3(a), which further accomplishes the stability of the three-phase system as discussed earlier. The subdomains P (0 < θ c < θ M , θ M for −ve γ) and Q (θ M < θ c < θ ∞, θ M for −ve γ) denote nucleation with the negative line tension effect, while the regions R (θ ∞ < θ c < θ ′), S (θ ′ < θ c < θ M , θ M for +ve γ), and T (θ M < θ c < 180◦, θ M for +ve γ) signify nucleation with a positive line tension. From Fig. 3(a), the following points can be clearly observed: (i) N is positive and D is negative in the subdomain P, (ii) both N and D are positive in the sections Q and R, (iii) N is negative and D is positive in the region S, and (iv) both N and D are negative in the subdomain T. So, the subdomains P and S are theoretically unstable for the negative line tension effect and positive line tension effect, respectively. While, Q (for the negative line tension), R, and T (for the positive line tension) are mathematically stable, as discussed earlier. Apart from the mathematical conclusions, some physical implications can also be drawn from the stability analysis for nucleation. Clearly, the variation in the microscopic contact angle is due to the effect of line tension. So, physically θ c should decrease with the increase in the negative line tension and it must increase with the increase in positive line tension. The intermolecular interactions in the vicinity of the triple contact line are responsible for this. However, the above mentioned physical consideration is violated in the subdomains P and T as shown in Fig. 3(b). So, region P is infeasible both mathematically and physically, although region T is mathematically feasible but physically unstable. Moreover, S is theoretically unstable as explained earlier. Nevertheless, the regions Q and R are analytically and substantially stable, and also, these two sections are basically the realm of interest for the present problem. The condition for the possible range of nucleation contact angles in simpler form can be rewritten as dF/dθ θ ′ for positive line tension as well as θ M < θ ∞ for negative line tension as considered in the preceding stability analysis.

III. RESULTS AND DISCUSSION

Section II illustrates a rigorous thermodynamic analysis based on CNT to discuss the heterogeneous nucleation on a planar substrate considering the line tension effect. In this section, first a graphical illustration of the generalized Young’s equation will be shown. It may be noted that θ c is a function of y and θ ∞ as interrelated through Eq. (21). However, all the three variables have a one-to-one correspondence under the extremum conditions. Therefore, variation of all the nondimensional parameters is described as a function of θ c and θ ∞ in this section for relevant consistency in the discussion. It may be seen from Sec. II that the shape factor, f , signifies the free energy barrier and can be considered as a function of θ c and θ ∞. Subsequently, we like to discuss the variation of the free energy barrier for nucleation and the effective peripheral, interfacial, and volumetric contribution on the work of formation for the heterogeneous nucleation.

FIG. 4. Variation of dimensionless line tension, y, with respect to θ c for different values of θ ∞.

A. Graphical interpretation of the generalized Young’s equation

Fig. 4 depicts the variation of y with respect to θ c for the entire range of θ ∞, covering both positive and negative line tensions. In all these curves, the variation is shown in the range of θ M < θ c < θ ∞ for negative line tension and θ ∞ < θ c < θ ′ for positive line tension. Eventually for each curve in Fig. 4, the upper and lower limits are constrained by the stability conditions for the effect of positive line tension and negative line tension, respectively, and these can be considered as the stability limits. Dimensionless line tension, y, is one of the principal parameters to control the variation of the contact angle, as discussed earlier. The three phase metastable system becomes unstable beyond these stability limits (θ c > θ M for y < 0 and θ c > θ ′ for y > 0) for a given θ ∞ and y, because near the vicinity of the triple line, the system does not possess excess energy due to molecular interactions sufficient to change the contact angle in the mentioned subdomains. Besides, the dotted curve (Y in Fig. 4) in between these limits can be represented by y = 0, and this curve basically depicts the change in sign of the line tension. For negative line tension, θ c decreases with | y | for a given θ ∞ in θ M < θ c < θ ∞. As discussed earlier (Fig. 2), negative line tension tends to spread the liquid in the vicinity of the boundary line resulting increment in the length of the triple contact line. On the other hand, θ c increases with | y | for a given θ ∞ in θ ∞ < θ c < θ ′ for positive line tension due to reduction in the length of the boundary line. Moreover, θ c increases with θ ∞ for a given | y | irrespective of the sign of line tension which further implies the strong dependence of the sign of line tension on the microscopic contact angle. In this context, considering the present geometric model for heterogeneous nucleation as a critical-value problem, it can be discerned that the equilibrium condition is obtained by extremizing Helmholtz free energy of formation, F(r, θ), with respect to the contact angle while the critical parameters are obtained by extremizing Gibbs free energy of formation, E(r, θ), with respect to the radius of the embryo. Individually,

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F has a local minimum for a given volume of the embryo at equilibrium condition and E has a local maximum for a given contact angle under critical condition. As can be observed from Fig. 3(b), θ c has got two numerical values for a given θ ∞ for a given lower value of | y |. These two values of θ c fall in the subdomains P and Q for negative line tension and the subdomains R and S or T for positive line tension as observed from Fig. 3(b). It can be shown that F(r c , θ c ) has a local minimum at Q for y < 0 and R for y > 0 and has a local maximum at P for y < 0 and S or T for y > 0. Therefore, it is true that the condition for equilibrium (Eq. (9) or (10)) is the necessary condition for the present three-phase metastable system. However, the condition for stability (Eq. (32)) is the sufficient condition for the present critical-value problem of heterogeneous nucleation. From our analytical model, it is found that the limiting value of y is | ylim| = 1.29, i.e., −1.29 ≤ y ≤ 1.29. When y is large, G and |γ| become higher, but r c has got lower values. Subsequently, the volume of the nucleus becomes very small. Beyond | ylim|, the volume of the embryo becomes smaller than the minimum volume of the critical nucleus for feasible heterogeneous nucleation and also F has got no extremum under such circumstances. B. Variation of the shape factor with respect to microscopic and macroscopic contact angle

Fig. 5 illustrates the variation of the shape factor f (θ c , θ ∞) with respect to θ c (θ ∞ < θ c < θ ′) for different values of θ ∞ and for positive line tension and negative line tension. In the absence of line tension effect, the factor always lies between 0 and 1. Under the action of line tension effect, the factor becomes greater than 1 for positive line tension and may assume a value less than zero for negative line tension in some cases. Fig. 5 also depicts that the extent and upper bound of f (θ c , θ ∞) decreases with the increase in θ ∞ for positive line tension. As θ c increases, the phenomenon tends towards dewetting, signifying detachment of the nucleus from the substrate—a condi-

FIG. 5. Variation of the shape factor f (θ c, θ ∞) with respect to θ c at different values of θ ∞.

J. Chem. Phys. 142, 104706 (2015)

tion conducive for homogeneous nucleation. Interestingly, at an extreme condition, θ ∞, θ c , and θ ′ become identical and coincide at a point (HoN) as shown in Fig. 5. Mathematically, at θ ∞ = 180◦, a condition similar to homogeneous nucleation will appear, though at this particular point, line tension has got no effect. Physically, however, the positive line tension effect creates the condition of homogeneous nucleation for a lower value of θ ∞ by contracting the triple line. HoN signifies the point of homogeneous nucleation. Another important feature to note from Fig. 5 is that as θ c increases, the interfacial area between the new stable phase and the nucleating agent decreases. As a result, the barrier height also increases as depicted in Fig. 5. Therefore, the geometric factor increases as critical microscopic contact angle increases for any fixed macroscopic contact angle. Moreover, for higher values of macroscopic contact angle, f assumes values greater than 1 for a certain range of critical microscopic contact angle. This suggests that, under such conditions, homogeneous nucleation is more favorable than heterogeneous nucleation, as the free energy barrier is higher for heterogeneous nucleation than that of the homogeneous nucleation for positive line tension effect. Definitely, the factor increases for lower values of θ ∞ for any fixed θ c due to increase in the effect of the line tension as discussed in Sec. III A. Essentially for each curve in Fig. 5, the upper and lower limits are restrained by the stability conditions for the effect of positive line tension and negative line tension, respectively, and the dotted curve (Y in Fig. 5) in between these limits is nothing but the transition curve from negative to positive line tension. Fig. 5 also illustrates the variation of the shape factor with respect to θ c and θ ∞, for the negative line tension effect. The minimum line tension for each θ ∞ occurs at θ M for negative line tension as depicted in Fig. 4 and Table I. However, f (θ c , θ ∞) becomes negative for lower values of θ c as can be observed from Fig. 5. This implies that the number of monomers is less than the number needed to form the critical nuclei and violates a basic assumption in the CNT. Therefore, essentially the theory requires a modification as proposed by Navascues and Tarazona.26 They have also suggested that the reason behind the negative free energy barrier might be due to the critical radius of the nucleus being comparable to atomic length scale or the negative line tension being outside the physical domain.26 There could be several reasons for this unique observation which poses a critical question to its feasibility. First, this barrierless nucleation is not uncommon in practice. In ioninduced nucleation, contribution of energy due to the electric field (which plays a role similar to that of negative line tension) reduces the barrier of free energy and even gives rise to a situation of barrierless nucleation. Further, fundamentally spherical-cap assumption may not be justified, because during the incipient period of nucleation, the nucleus is atomistically small and θ in Fig. 1 must be considered as the microscopic contact angle for the thermodynamic analysis. For contact angle less than 90◦, negative line tension may also promote the formation of a precursor or thin film31,32 (Fig. 2(b)). These films may lead to an increase in free energy barrier as the formation and subsequent growth of new embryo occurs on these films of same phase. Such an eventuality is

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not amenable within the framework of the present model. However, it needs to be kept in mind that the physics of triple line is extremely complex and is not yet fully understood. The transition from a discrete atomic picture to the mesoscopic continuum is neither captured in the present model nor explicitly treated by other existing models. The above postulates therefore cannot be substantiated without further research. However, the shape factor increases with the increase in θ c and with the decrease in θ ∞ in the range θ M < θ c < θ ∞ as shown in Figure 5. For any fixed θ ∞, the negative line tension effect increases with the decrease in θ c . This is due to an enhanced wetting and subsequent increase in the boundary line length. Nevertheless, for a given θ c , an increase in θ ∞ demands a higher level of intermolecular interactions at the triple line to spread it out further. One can appreciate from Fig. 5 that the nucleation barrier height decreases considerably with the algebraic value of the negative line tension. Moreover, irrespective of θ c , the free energy barrier for nucleation increases for higher values of θ ∞. C. Comparison among the peripheral, interfacial, and volumetric contribution of the shape factor

For a better understanding of the line tension effect,11 it would be appropriate to consider the contribution of the three factors of the shape function, f (θ c , θ ∞). After substituting Eq. (21) in Eq. (20), the factor can be rearranged separately as follows:   3 2 f = sin θ c (cos θ ∞ − cos θ c ) 2   3 + 2 (1 − cos θ c ) − cos θ ∞sin2θ c 4  
1 3 2 − 3 cos θ c + cos θ c . (33) − 2 The peripheral ( f L ), interfacial ( f S ), and volumetric ( f V ) contribution of the shape factor can be easily identified, where 3 2 sin θ c (cos θ ∞ − cos θ c ) , 2

(34)

3 2 (1 − cos θ c ) − cos θ ∞ sin2θ c , 4

(35)

f L (θ c , θ ∞) = f S (θ c , θ ∞) = and


1 2 − 3 cos θ c + cos3θ c . (36) 2 Based on the above rearrangement of the shape factor, Fig. 6 illustrates the contribution of the peripheral and interfacial effects for positive line tension and negative line tension with respect to θ c for different values of θ ∞. Interestingly, f L is always negative for negative line tension and is positive for positive line tension. From the figure, it can be clearly discerned that for the possible nucleation contact angle, both the contributions increase with the increase in θ c . However, for a fixed θ c , as θ ∞ increases, f L decreases but
f S increases. This restricts the contribution of f H = f L + f S within the range (0, 3.2) for positive line tension and (−0.4, 3.0) for negative line tension as can be seen from Fig. 7(a). Basically, f H is the non-dimensional critical Helmholtz free energy of formation f V (θ c , θ ∞) = −

FIG. 6. Contribution of the effect of line tension, (a) f L and surface tension, (b) f S in the shape factor ( f ) as a function of θ c for different values of θ ∞.

of a spherical-cap shaped nucleus. The contribution f V , on the other hand, does not change with the change in the nature of the line tension. It is always in the range (−2.0, 0) and it decreases with an increase in θ c (Fig. 7(b)). As a result, the overall shape factor is in the range of (0, 1.7) for positive line tension and of (−0.7, 1.0) for negative line tension as can be observed from Fig. 5. Fig. 8 shows a quantitative comparison between f L and S f . For some smaller values of macroscopic contact angle as well as microscopic contact angle, f L becomes higher than f S . As the substrate becomes more wettable for lower values of contact angle and subsequently the length of the triple line increases, line tension effect prevails over surface tension effect for lower values of contact angle. D. Quantitative assessment of the present analysis

The analysis of heterogeneous nucleation in the presence of line tension based on a rigorous thermodynamical

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FIG. 7. (a) Cumulative contribution due to the effect of line tension and surface tension, f L + f S (= f H ), and (b) volumetric contribution, f V , of the shape factor ( f ) as a function of θ c and θ ∞.

FIG. 8. Comparison of f L and f S in the shape factor ( f ) as a function of θ c and θ ∞.

formulation has been presented. Based on the results obtained, purely from a mathematical point of view, one can identify the regime of nucleation for different thermodynamical and physicochemical properties (expressed nondimensionally) and wetting characteristics (in terms of contact angle) of the fluid, covering both negative line tension and positive line tension. These have been presented in Secs. III A–III C. A question

still remains how far this mathematical analysis is relevant for real fluids with typical property values. Particularly, the present work focuses on the effect of line tension whose magnitude not only lies within a range but also is extremely small. It, therefore, will be prudent to consider the nucleation data of certain real fluids and to do the estimation based on the present analysis. Two types of phase transformation, i.e., condensation

TABLE II. Physicochemical properties and critical nucleation parameters of some alkanes, water, and heavy water during condensation. S denotes supersaturation ratio, defined as p/p sat .

Liquid

T (K)

ρ liq (1027 m−3)

G/lnS (107 N/m2)

σ (10−2 N/m)

r c × lnS (10−9 m)

γ lim × lnS (10−11 N)

δ lim × lnS (10−9 m)

Water33,34 D2O33,34 Ethane33,35 n-butane33,35 n-heptane33,35

300 298 155 216 248.75

33.422 33.215 11.609 6.823 4.331

13.8430 13.6656 2.4843 2.0348 1.4870

7.174 7.187 2.090 2.180 2.480

1.0365 1.0518 1.6826 2.1427 3.3350

9.590 9.750 4.536 6.026 10.669

1.3371 1.3568 2.1706 2.7641 4.3022

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TABLE III. Physicochemical properties and critical nucleation parameters of water and hexadecane during crystallization. ∆S m is the entropy of fusion in J/m3 K and ∆T the supercooling in K. Liquid Water36,37 Hexadecane33,38

G/∆T (= ∆S m ) (105 N/m2 K)

σ (10−2 N/m)

r c × ∆T (10−8 m K)

γ lim × ∆T (10−11 N K)

δ lim × ∆T (10−8 m K)

12.2101 6.0222

3.0 0.387

4.914 1.285

1.9 6.4

6.339 1.658

(G = k BT ρliq × ln S, k B is the Boltzmann constant, ρliq is the number density of liquid, and T is the temperature) from a supersaturated vapor and crystallization (G = ∆Sm · ∆T) from a supercooled liquid, have been considered. As the measured value of the line tension pertaining to the nucleation process is rarely available, we consider an indirect approach. It is known that the typical line tension value lies between 10−10 N and 10−12 N.16–20,27 Using the nucleation data, we have predicted line tension based on our analysis and confirmed that it lies within the viable range. The critical radius (r c ) of nucleation is obtained using the thermodynamical and physicochemical properties of the different fluids and the limiting values of nondimensional line tension ( ylim) have been considered 1.29 and −1.29 for positive line tension and negative line tension as depicted in Fig. 4. Parallelly, maximum or limiting values of positive line tension and negative line tension (γlim) are calculated with the help of the following equation, i.e., γlim = ylim × σ × r c (last column of Tables II and III). Table II depicts the estimation during condensation, while Table III presents values for crystallization. Interestingly, it can be observed that these limiting values of positive line tension and negative line tension are supersaturation dependent for condensation and supercooling dependent for crystallization. Therefore, an important finding from our thermodynamic model is that the line tension is definitely a function of metastability in case of heterogeneous nucleation. Nevertheless, the limiting values estimated for the line tension are found to be of the same order of magnitude typically reported in the literatures. In general, the line tension in heterogeneous nucleation is not only a function of the degree of metastability but also dependent on the apparent and Young’s contact angle. E. Comparison with experimental data

Our formulation and the analysis there after clearly show that line tension has a significant effect on the process of heterogeneous nucleation. The experimental verification could be challenging due to the extremely small size of the nucleus, substantially low value of line tension, and the difficulty of getting a single nucleus on any substrate. On the other hand, nucleation experiments can reveal the kinetics of nucleation from which an indirect assessment of the thermodynamics of the process may be attempted. In this present analysis, we have considered a dimensionless number y to include the effect of line tension in the thermodynamical formulation. The nondimensional number includes the peripheral tension, the critical radius, and the interfacial tension between the condensed phase and the metastable phase. In other words, y is a function of the line tension (γ), the surface tension (σ), and the chemical potential difference (G) between the metastable phase and the condensed

phase per unit molecular volume of the condensed phase of the metastable system. In this context, Heneghan et al. discussed the kinetics of heterogeneous ice nucleation on a glass substrate for supercooled pure water and AgI-seeded water using the automated lag-time apparatus (ALTA).37 Though the authors did not report the value of the critical radius in their experiment, the same can be estimated from their reported data. The average lag-time τ, reported in the work, is basically the reciprocal of the nucleation rate, J, i.e., τ = 1/J. From the theory of nucleation kinetics, the rate of heterogeneous nucleation can be expressed as J = J0 exp (−Ec /k BT) = J0
exp −Echomo · W/k BT , where k B is the Boltzmann constant, J0 is the pre-exponential factor, and W is the shape factor which takes care of the line tension. Further, average ( lag-time of ) nucleation can be modified as τ = (1/J0) · exp z · W/T(∆T)2 , where z = (16π/3k B) σ 3(Tm/∆Hm )2. Experimentally,37 it is found that J0 = 1.78 × 10−10 s−1, W = 5.6066 × 10−4, and z = 19.886 × 106 K3 for pure water. Now, for heterogeneous ice nucleation, G can be written in terms of supercooling as G = (∆Hm/Tm )∆T, where ∆T = (Tm − T) is the supercooling in K, ∆Hm is the latent heat of fusion per unit molecular volume in J/mol, and Tm is the freezing temperature in K. This gives G = 1.2836 × 106 × ∆T J/m3. The critical radius for heterogeneous nucleation in the reported experiment can be readily obtained as r c = 2σ/G = (4.6743/∆T) × 10−8 m, taking σ = 3 × 10−2 N/m as ice-water interfacial tension. From our theoretical analysis, it has been found that y always lies within (−1.29, 1.29) . Moreover, substituting the typical value of the line tension of the order of 10−10 N and the typical surface tension of the order of 10−2 N/m in the expression of the critical radius from the thermodynamical formulation, i.e., r c = γ/σ ylim, we get r c is of the order of 10−8 m, which further validates the experimental observations. So, one can easily understand the effect of the variation in each individual parameters (γ, σ, G or γ, σ, r c ) on nucleation based on the variation in a single dimensionless parameter, y, without considering the details of the individual parameters.

IV. CONCLUSIONS

We have formulated a modified geometric model for the critical-value problem of heterogeneous nucleation on a planar surface including the line tension effect based on a systematic and rigorous thermodynamic formulation. The geometric factor or the shape factor, f (θ c , θ ∞), raised in the formulation, is the ratio of the reversible work to form a nucleus for heterogeneous nucleation with line tension effect to that of the homogeneous nucleation. Many of the unique traits of heterogeneous nucleation in the presence of line tension effect can be explained in terms of f . From our analysis, we draw the following conclusions:

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(1) The possible range of critical microscopic contact angle, which was obtained by combining the results from the generalized Young’s equation and from the stability analysis, is θ ∞ < θ c < θ ′ for positive line tension and is θ M < θ c < θ ∞ for negative line tension. Seemingly, an earlier presumed appreciable range for θ c (0 < θ c < θ ∞) is found not to be true for the negative line tension effect. (2) The effect of positive line tension increases with θ c and decreases with θ ∞ in θ ∞ < θ c < θ ′. Similarly, the shape factor becomes higher considerably for lower values of θ ∞ and higher values of θ c for positive line tension. Interestingly, the occurrence of heterogeneous nucleation is less favorable than homogeneous nucleation for higher y, lower θ ∞, or higher θ c . (3) As the contact angle decreases, the substrate becomes more wettable and the length of the triple line increases. As a result, the contribution of f L in f becomes more dominant compared to the contribution of f S . The value of ( f L + f S ) is bound within (0, 3.2), whereas the volumetric contribution f V is always in (−2.0, 0), resulting the overall shape factor f is in the range of (0, 1.7) for positive line tension. (4) The important parameters derived in the present formulation are the length scales, δ = σγ , r c = 2σ G , and the dimensionless line tension, y = rδc . If y is very small, then θ c  θ ∞ from Eq. (21), and the locus of shape factor becomes very close to the transition curve ( y = 0) of Fig. 5. Then, the negative line tension does not necessarily lead to a negative barrier. Nearly 30% reduction of θ c is required to make the free energy barrier negative from Fig. 5. It is observed from our analysis that the limiting value of line tension in heterogeneous nucleation is dependent on supersaturation for condensation and on supercooling for crystallization. Further, the line tension is not only a function of the degree of metastability but also dependent on the microscopic and macroscopic contact angle. A semiquantitative analysis has been made using the property values of the real fluids and the probable magnitude of line tension. The results are in agreement with the present analysis. The value of critical radius during heterogeneous nucleation in the presence of line tension predicted by our theory matches reasonably well with the data of a typical nucleation experiment.

where F ′ = F − λV and λ is the Lagrange multiplier. Substituting the value of F ′ from Eq. (8) in Eqs. (A1) and (A2), one can easily derive  2σr 2 (1 − cos θ) − cos θ ∞sin2θ + 2γ sin θ λ= r 2 (2 − 3 cos θ + cos3θ) (A3) and 2σr {sin θ − cos θ ∞ sin θ cos θ} + 2γ cos θ . r 2sin3θ

λ=

(A4) Therefore by equating both λ from Eqs. (A3) and (A4), the generalized Young’s equation, i.e., Eq. (9) or (10), can be easily derived.

APPENDIX B: EVALUATION OF dθ/dr

Differentiating Eq. (10) with respect to θ, we get δ dr · = cos θ (cos θ ∞ − cos θ) + sin2θ, r 2 dθ and after rearrangement, we have −

dθ δ/r =−
. dr r cos θ ∞ cos θ − cos2θ + sin2θ

(B1)

(B2)

Substituting value of δ/r from Eq. (10) in the above equation, Eq. (14) can be obtained.

APPENDIX C: CALCULATION OF PARTIAL DERIVATIVES OF F AND F ′′ WITH RESPECT TO r AND θ

Using Eq. (12), we get 
∂F = σ 4πr (1 − cos θ) − cos θ ∞ × 2πrsin2θ ∂r + (γ × 2π sin θ)  = 2πσr × 2 (1 − cos θ) − cos θ ∞ sin2θ + 2πσr sin2θ (cos θ ∞ − cos θ)
= 2πσr 2 − 3 cos θ + cos3θ

(C1)

and 
∂F = σ 2πr 2 sin θ − cos θ ∞ × 2πr 2 sin θ cos θ ∂θ + (γ × 2πr cos θ) = 2πσr 2 × {sin θ − cos θ ∞ sin θ cos θ} + 2πσr 2 sin θ cos θ (cos θ ∞ − cos θ)

ACKNOWLEDGMENTS

The authors appreciate the anonymous referee for thoughtful comments and suggestions for the improvement of the manuscript.

= 2πσr 2sin3θ. APPENDIX A: DERIVATION OF GENERALIZED YOUNG’S EQUATION

(C2)

Similarly from Eq. (13), we have
∂F ′′ = πGr 2 2 − 3 cos θ + cos3θ ∂r

The Lagrange multiplier technique demands ∂F ′ =0 ∂r

(A1)

∂F ′ = 0, ∂θ

(A2)

and

(C3)

and ∂F ′′ = πGr 3sin3θ. (C4) ∂θ Substituting Eqs. (C1)–(C4) in Eq. (15), Eq. (16) can be easily obtained.

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where Fc is the critical Helmholtz free energy and y (=δ/r c ) is the dimensionless line tension. From the stability analysis, using Eq. (32), we have

APPENDIX D: DERIVATION OF EQ. (32)

Equation (B2) can be rewritten as   (δ/r) · cos θ + sin3θ dr = −r . (δ/r) sin θ dθ

dF < 0. d ln y r =r c,θ=θ c

(D1)

As F is function of r and θ, therefore, dF ∂F ∂F dr = + · . (D2) dθ ∂θ ∂r dθ Substituting Eqs. (C1), (C2), and (D1) in Eq. (D2), we find 
dF 2 = 2πσr sin3θ − 2 − 3 cos θ + cos3θ dθ   (δ/r) · cos θ + sin3θ · (δ/r) sin θ 2  2πσr (δ/r) sin4θ − (2 + cos θ) (1 − cos θ)2 = (δ/r) sin θ   · (δ/r) · cos θ + sin3θ =

 2πσr 2(1 − cos θ)2  · (δ/r) − sin3θ (2 + cos θ) . (δ/r) sin θ

(D3)

From Eq. (10), we have 1 d (δ/r) d ln (δ/r) = (δ/r) dθ dθ 1  = cos θ (cos θ ∞ − cos θ) + sin2θ (δ/r)  1 = · (δ/r) · cos θ + sin3θ . (D4) (δ/r) sin θ Using Eqs. (D3) and (D4), one can obtain dF dF/dθ = d ln (δ/r) d ln (δ/r) /dθ = =

2πσr 2(1−cos θ)2  · (δ/r) − sin3θ (2 + cos θ) (δ/r ) sin θ  3 1 (δ/r ) sin θ · (δ/r) · cos θ + sin θ  (δ/r) − sin3θ (2 + cos θ) 2 2 2πσr (1 − cos θ) ·  . 3 (δ/r) · cos θ + sin θ

(D5) Therefore, for the present critical-value problem, Eq. (D5) can be rewritten as   3 dFc 2 y − sin θ c (2 + cos θ c ) 2 = 2πσr c (1 − cos θ c ) , d ln y y cos θ c + sin3θ c (D6)

1D.

(D7)

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