THE JOURNAL OF CHEMICAL PHYSICS 143, 204703 (2015)

Thermokinetics of heterogeneous droplet nucleation on conicallytextured substrates

Sanat K. Singha, Prasanta K. Das,a) and Biswajit MaitiDepartment of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

(Received 7 July 2015; accepted 5 November 2015; published online 24 November 2015)

Within the framework of the classical theory of heterogeneous nucleation, a thermokinetic modelis developed for line-tension-associated droplet nucleation on conical textures considering growthor shrinkage of the formed cluster due to both interfacial and peripheral monomer exchange and byconsidering different geometric configurations. Along with the principle of free energy extremization,Katz kinetic approach has been employed to study the effect of substrate conicity and wettabilityon the thermokinetics of heterogeneous water droplet nucleation. Not only the peripheral tension isfound to have a considerable effect on the free energy barrier but also the substrate hydrophobicity andhydrophilicity are observed to switch over their roles between conical crest and trough for differentgrowth rates of the droplet. Besides, the rate of nucleation increases and further promotes nucleationfor negative peripheral tension as it diminishes the free energy barrier appreciably. Moreover,nucleation inhibition can be achievable for positive peripheral tension due to the enhancement ofthe free energy barrier. Analyzing all possible geometric configurations, the hydrophilic narrowerconical cavity is found to be the most preferred nucleation site. These findings suggest a physicalinsight into the context of surface engineering for the promotion or the suppression of nucleation onreal or engineered substrates. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4935953]

I. INTRODUCTION

Nucleation can be defined as the mechanism of theformation of new stable phase in a form of cluster from ametastable parent phase. Nucleation can be homogeneous orheterogeneous in nature.1 Heterogeneous nucleation is notonly more common but also has important applications inmaterials and biology.2 In spite of many controversies onthe associated postulations, the classical nucleation theory(CNT) is one of the most espoused approaches for elucidatingthe heterogeneous nucleation and has been extensivelyemployed for various substrate geometries. Heterogeneousnucleation on a planar3 or spherical4–7 substrate is mostfrequently encountered in the literature. However, circularconical textures, e.g., crest and trough, are omnipresentas defects in the actual substrates and also imitate themicrostructures of some typical plant leaves. Therefore, forheterogeneous nucleation, the cone-shaped textured substratecan be considered as a generic geometry, although the topic isrelatively less studied in the literature.

Line tension or peripheral tension in connection with thewetting in a three-phase system, first conceived by Gibbs8

around hundred years back, continues to intrigue the scientificcommunity regarding its value and sign.9 Line tension cannotbe disregarded during the initial period of the heterogeneousnucleation, i.e., when the newly formed cluster is small-sizedin molecular length scale. Taking our cue from Gretz,10

the inclusion of line tension in the classical theory for thethermodynamic model of heterogeneous nucleation on the

a)Author to whom correspondence should be addressed. Electronic mail:pkd@mech.iitkgp.ernet.in.

substrates of different geometries has been inspected fromtime to time.11–16

The kinetic analysis, based on the classical theory ofheterogeneous nucleation, is also an expedient method toinvestigate nucleation phenomena as discussed earlier. Inthis regard, it can be mentioned that the kinetic model forhomogeneous nucleation has been extensively used, as acomparatively complete theory on the kinetic analysis forhomogenous nucleation is available in the literatures.1,2,17,18

Some kinetic models have been developed to reveal thecharacteristics of the heterogeneous nucleation on thesubstrates of various geometries, e.g., planar,19 spherical,20

conical,21 etc. However, the kinetics of the heterogeneousnucleation is still not clearly understood due to the intricateinfluence of the wettability and roughness of the substrate.In the kinetic model for heterogeneous nucleation, theattachment of monomers to the cluster can take place via twoprocesses:22–25 (i) explicit interfacial accumulation from themetastable phase at the interface of the cluster and the parentphase and (ii) implicit peripheral addition through the vicinityof the triple line from the adsorbed monomers on the substrate.Recently, Luo et al.25 developed a novel kinetic model forcondensation or droplet nucleation on a spherical substrate byconsidering the combined effect of the line tension and the twoprocesses of monomer attachment on nucleation associatedwith condensation. However, a similar thermokinetic modelfor heterogeneous nucleation on a conically textured substratehas yet to be developed.

As discussed earlier, the effect of triple line cannotbe ignored during the primordial stage of the clusterevolution. So, in the present paper, within the frameworkof CNT, we discuss the thermokinetic characteristics of

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204703-2 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

conical-texture-mediated condensation by considering thecollective influence of the line tension and the twoprocesses of monomer exchange on nucleation for differentgeometric configurations. The physical model of conical-texture-mediated condensation is developed in Sec. II. InSec. III, a rigorous energetic model for line-tension-associatedcondensation is studied and the associated critical parametersare derived. Taking water as the test fluid, a kinetic model fordroplet nucleation on a conically textured substrate is analyzedand the related rate coefficients are obtained in Sec. IV. Sec. Vdiscusses about the control of nucleation and the preferrednucleation sites for different configurations. A brief discussionon the results obtained from the present thermokinetic modelis summarized in Sec. VI.

II. GEOMETRIC MODEL

Let us consider a spherical-shaped embryo grows witha wetting angle θ on conical textures, the cone angles areφ and φ∗ (φ∗ = 180◦ − φ) for crest and trough, respectively.Fig. 1 illustrates the three configurations each for conicalcrest and trough. The configurations depend on the distancebetween the apex of the cone and the spherical center ofthe formed droplet. Moreover, the mentioned distance is afunction of the wettability and the conicity. In case of conicalcrest, if θ ≥ 90◦ + φ (case A), the cluster could not be ableto grow on the extrinsic substrate (homogeneous nucleation)due to the extremely unfavorable nucleation condition. If,however, 90◦ − φ∗ < θ < 90◦ + φ∗ (case B), nucleation occursat the apex of the cone. Finally, if θ ≤ 90◦ − φ∗ (case C),then the favorable nucleation site is away from the cone apex.Interestingly, the conical substrate becomes flat (case D) whenφ = φ∗ = 90◦, which is one of the limiting cases of the presentphysical model. In case of conical cavity, if θ ≥ 90◦ + φ∗

(case E), i.e., a large wetting angle and a narrow cone, thepreferred nucleation spot is away from the cone apex. If,however, 90◦ − φ∗ < θ < 90◦ + φ∗ (case F), nucleation occursat the apex of the cone.26,27 Finally, if θ ≤ 90◦ − φ∗ (case G),

FIG. 1. Various possible configurations of conical-texture-mediated nucle-ation as a function of θ and φ or φ∗.

FIG. 2. Illustration of formation of a droplet (blue) on different conicalsurface textures (red): (a) Crest or vertex and (b) trough or cavity.

then the new phase can grow from the cone apex withoutan energy barrier, i.e., non-activated barrierless nucleation.Therefore, cases A, D, and G are the limiting cases of thepresent physical model and the thermokinetic aspects of casesB, C, E, and F will be discussed thoroughly in Secs. III–VI.

The complete geometric models for cases B and F areshown in Fig. 2. Consider the formation of a spherical-capcluster or droplet of radius r on a conically textured substrateof half-crest angle, φ or half-trough angle, φ∗ as illustratedin Figs. 2(a) and 2(b), respectively. A droplet (β) is emergedat the interface between the supersaturated vapor phase (α)and the substrate (N). O is the spherical center of β. S,which is basically the projected point of triple line, is a jointwhere the three phases α, β, and N encounter each other. θis the microscopic wetting angle between β and N . Similarly,complete geometric model can be obtained for cases C and E.

Various associated geometric parameters of cases B, C,E, and F are shown in Table I in which L is the length of theperiphery of the triple line, V is the truncated volume of thecluster or the droplet and the interfacial areas between α andβ and that between β and N can be given by Aαβ and AβN ,respectively.

III. ENERGETIC MODEL

A. Minimization of Helmholtz free energy change

For obtaining a mathematical relationship between thesubstrate wettability and the line tension, the extremization ofthe Helmholtz free energy is needed. The change of Helmholtzfree energy (∆FH) associated with the formation of a clusterfrom the supersaturated vapor on an extrinsic substrate can beexpressed as

∆FH = σαβAαβ +�σβN − σαN

�AβN + γL, (1)

where γ is the line tension and σαβ, σαN , and σβN are thesurface tensions between droplet–vapor, droplet–substrate,and vapor–substrate, respectively.

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204703-3 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

TABLE I. Geometric parameters for various possible configurations of conical-texture-mediated nucleation.

Geometry L2πr

Aαβ

4πr2

AβN

4πr2 V /(

43πr

3) (= f )

Case B (crest) cos(θ−φ) 1+sin(θ−φ)2

cos2(θ−φ)4sinφ

14

(2+3sinψ−sin3ψ−cos3ψ · cosφ

sinφ

),

ψ = θ−φCase C (crest) 2cosθcosφ sinθcosφ cosθ sinθcosφ sin3θcosφCase E (trough) −2cosθcosφ∗ 1−sinθcosφ∗ −cosθ sinθcosφ∗ 1−sin3θcosφ∗

Case F (trough) −cos(θ+φ∗) 1−sin(θ+φ∗)2

cos2(θ+φ∗)4sinφ

14

(2−3sinψ+sin3ψ−cos3ψ · cosφ∗

sinφ∗),

ψ = θ+φ∗

Now from the well-known Young equation, the macro-scopic wetting angle (θ∞) can be defined as

cos θ∞ =σαN − σβN

σαβ. (2)

In this context, it can be found that only the interfacialtension σαβ remains in the subsequent analyses, because thedifference of the other two can be expressed in terms of σαβand θ∞ with the help of Eq. (6). So, for brevity, we willconsider σαβ as σ in the rest of the paper.

As the change of Helmholtz free energy, associated withthe formation of a droplet, has a local minimum for a givenvolume of the droplet, the equilibrium condition can beobtained by extremizing ∆FH with respect to the wettingangle. Therefore, the minimization of ∆FH can be achievedusing the Lagrange multiplier technique15 (Appendix A) underthe supplementary condition of constant volume. The resultingcondition for equilibrium for cases B and F is found to be as

γ

σr=

cos (θ − φ)sin φ

· (cos θ∞ − cos θ) , for case B, (3a)

= −cos (θ + φ∗)sin φ∗

· (cos θ∞ − cos θ) , for case F. (3b)

However, the equilibrium condition becomes independentof φ for cases C and E and can be expressed as

γ

σr= sin θ (cos θ∞ − cos θ) (3c)

and it resembles the corresponding condition for planar-substrate-induced nucleation.

Eq. (3) is also known as the modified Young’s equationor modified Dupre-Young equation. Though the analysis isformulated considering line tension as positive, however, nothermodynamical theory is violated if one considers negativeline tension in the analytical derivation. Fig. 3 illustrates thevariation of the microscopic contact angle with the sign ofthe line tension. From Eq. (3), the microscopic contact angleis found to be greater than macroscopic contact angle forpositive value of line tension, resulting the constriction of thetriple line. Moreover, θ becomes smaller than θ∞ for negativeline tension which further suggests that the elongation of thetriple line. This is similar to the planar case for cases C and E(Figs. 3(b) and 3(d)) as mentioned earlier, though θ dependsnot only on the line tension but also on the conicity for casesB and F (Figs. 3(a) and (3(c)). Moreover, from Table I, itcan be found that there are two triple lines associated withcases C and E. Basically, the formation of the embryo firsthomogeneously formed from the metastable vapor phase thenits subsequent evolution starts on the slant interfacial areas ofthe conical structures for cases C and E.

B. Maximization of Gibbs free energy change

To obtain the critical parameters of heterogeneousnucleation, maximization of the change in Gibbs free energywith respect to the size (r or n) of the cluster is needed. The

FIG. 3. Variation of microscopic wet-ting angle with respect to the sign ofline tension: (a) case B, (b) case C,(c) case F, (d) case E (+ve line ten-sion (green), −ve line tension (red)).An exaggerated view of vicinity of thetriple line for various cases is shown in(I)–(VI).

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204703-4 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

Gibbs free energy change related to the formation of a clusteris

∆G = ∆FH − ∆FV , (4)

where ∆FH and ∆FV is the variation in the Helmholtz andvolumetric free energy due to the formation of a cluster,respectively.

The volumetric free energy change (∆FV) is associatedwith the variation in the free energy related to the differencein pressure (∆p) between the formed droplet and thesupersaturated vapor. ∆FV can be expressed as

∆FV = V∆p, (5)

where

∆p =kBTvm· lnS, (6)

kB is the Boltzmann constant, T is the temperature, and S isthe supersaturation.

To determine the critical free energy barrier, one needs∂∆G/∂r |r=rc = 0 or ∂∆G/∂g |n=nc = 0.

After differentiating Eq. (4) with respect to r , we have

∂∆G∂r=∂∆FH

∂r− ∂∆FV

∂r. (7)

Substituting Eqs. (B1)-(B4) and (B6)-(B9) fromAppendix B in Eq. (7), we get

∂∆G∂r= 4πr2

(2σr− ∆p

)f . (8)

Further, using Eqs. (1) and (14), we have

∂∆G∂n=∂∆G∂r· ∂r∂n= Vm

(2σr− ∆p

). (9)

So, from extremization principle, one gets the criticalradius as

rc =2σ∆p

. (10)

Moreover, the critical number of molecules within theformed stable cluster can also be given by

nc =V (rc)

Vm. (11)

This result clearly indicates that the critical size of thedroplet depends neither on the substrate conicity (φ) noron the line tension (γ) and it has the same expressionfor the present modified case and for the other limitingcases, i.e., homogeneous condensation and heterogeneous

condensation on planar substrates without considering theeffect of line tension.

The volume of the droplet can be rewritten as

V = nVm =4π3

r3 f (12)

and using Eq. (12), one can get

∂r∂n=

Vm

4πr2 f. (13)

The second differential of ∆G with respect to n can beevaluated with the help of Eqs. (9) and (13) as

∂2∆G∂n2 =

∂

∂r

(∂∆G∂n

)· ∂r∂n= − σVm

2

2πr4 f< 0, (14)

which in turn fulfills the condition of maximality.Substituting the expression of the critical radius in the

expression of the Gibbs free energy change, one can determinethe free energy barrier. From the expression of the energybarrier, one can easily identify the component without theinfluence of line tension and the part due to the line tension,where

∆Gc = ∆Gcgeom + ∆Gc

peri

= ∆Gchom f (θc, φ) + πrcγg (θc, φ) , (15)

where

∆Gchom =

43πσrc2, (16)

and f and g can be termed as the geometric factor and theperipheral factor, respectively.

The values of f and g as function of geometric parametersare shown in Table II for different configurations. While thegeometric factor is directly associated with the truncatedvolume of the cluster and the peripheral factor is explicitlyrelated to the line-tension-associated wetting. From Table II,the geometric factor and the peripheral factor are found to beproportional to the volume of the cluster and the length of thetriple line, respectively.

In the paradigm of the present extended model, let usconsider four limiting case. First, when the influence of theline tension is ignored, the expression for the reversible workof formation of a droplet for the case of the heterogeneouscondensation on a conical substrate can be derived fromEq. (15) as21

∆Gc (θc = θ∞, φ,γ = 0) = 4πσr2c

3× f . (17)

TABLE II. Geometric and peripheral factors for various configurations.

Geometry µ(=

γσr (cosθ∞−cosθ)

)f

(=∆Gc

geom

∆Gchom =

∆Gcgeom

(4/3)πσrc2

)g

(=∆Gc

peri

πrcγ= L

πrc− AβN

µ ·πrc2

)Case B (crest) cos(θ−φ)

sinφ14

(2+3sinψ−sin3ψ−cos3ψ · cosφ

sinφ

),

ψ = θ−φcos(θ−φ)

Case C (crest) sinθ sin3θcosφ 0Case E (trough) sinθ 1−sin3θcosφ∗ 0Case F (trough) − cos(θ+φ∗)

sinφ∗14

(2−3sinψ+sin3ψ−cos3ψ · cosφ∗

sinφ∗),

ψ = θ+φ∗−cos(θ+φ∗)

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204703-5 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

Then, considering the effect of the line tension, theexpression of the free energy barrier for the case of theheterogeneous condensation on a planar substrate can beobtained from Eq. (15) as10,11

∆Gc (θc, φ = φ∗ = 90◦)=

16πσ3

3(∆p)2 ·

14·�2 − 3 cos θc + cos3θc

�+

2πγσ∆p

· sin θc

. (18)

Further, when the influence of the line tension isignored, the well-known expression for the reversible workof formation of a droplet for the case of the heterogeneouscondensation on a flat surface can be derived from Eq. (15)as1,2

∆Gc (θc = θ∞, φ = φ∗ = 90◦, γ = 0)=

16πσ3

3(∆p)2 ·

14·�2 − 3 cos θ∞ + cos3θ∞

�. (19)

FIG. 4. Depiction of the geometric factor, f for a conical crest (a) and trough(b).

Finally, the well-known expression for the free energybarrier for homogeneous condensation can be obtained fromEq. (15) as1,2

∆Gc (θc = 180◦, φ = φ∗ = 90◦, γ = 0)= ∆Gc

hom =4πσr2

c

3=

16πσ3

3(∆p)2 . (20)

Fig. 4 shows the variation of the geometric factor asa function of θ and φ or φ∗ for conical crest and trough,respectively. Basically, the geometric factor can be consideredas a dimensionless free energy barrier without considering theinfluence of line tension. Irrespective of the configurations,the geometric factor is found to have lower values for lowersubstrate wettability and higher conicity for crest and lowerconicity for trough as can be observed from Fig. 4. Therefore,nucleation can be promoted for higher φ or lower φ∗ withlower wettability in the absence of line tension. Moreover,nucleation hindrance is possible for lower θ and lower φ for

FIG. 5. Illustration of the peripheral factor, g for a conical crest (a) andtrough (b).

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204703-6 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

TABLE III. Parameters used for evaluation of various rate coefficients asso-ciated with the kinetic model of heterogeneous condensation on a conicallytextured substrate.

Parameter Symbol (unit) Value

Physicalconstant

Boltzmannconstant

kB (J K−1) 1.381×10−23

Surroundingcondition

Supersaturation S 2.0Temperature T (K) 300.0

Waterproperty28,29

Molecularvolume

Vm (m3 molecule−1) 2.992×10−29

Molecular mass mm (kg molecule−1) 2.982×10−26

Saturation vaporpressure

psatv (N m−2) 3570.0

Interfacialproperty30

Surface tensionof water

σ (N m−1) 7.174×10−2

Kineticparameter24

Stickingcoefficient

α 1.0

Averagejumpingdistance

δ (m) 3.2×10−10

Adsorptionenergy

∆Gads (J molecule−1) 2.9×10−20

Diffusionenergy

∆Gdiff (J molecule−1) 2.9×10−21

conical asperity or higher φ∗ for conical cavity without theeffect of line tension.

Fig. 5 depicts the variation of the peripheral factor, gas a function of the substrate wettability and conicity forboth the conical asperity and cavity. From Tables II and III,it can be found that the peripheral factor is proportionalto the length of the triple line and therefore, it variesaccordingly for cases B and F. So, the trend of the curvesin Figs. 5 and 6 is similar in nature for cases B and F.The factor has lower values in the hydrophobic regimefor conical crest and in the hydrophilic regime for conicaltrough. Moreover, there exists a maximum at θ = φ (case B)and θ = 180◦ − φ∗ (case F) for conical asperity and cavity,respectively, for cone-angle greater than 90◦. Interestingly,there is no peripheral component of the energy barrier forcases C and E as can be seen from Fig. 5. Moreover, fromTable III, it is observed that the peripheral contribution ofthe free energy barrier becomes zero for cases C and E asdiscussed earlier.

IV. KINETIC MODEL

A. Attachment and detachment rates

In the preceding thermodynamic analysis of the hetero-geneous condensation, we are concerned neither in the detailsof the droplet formation nor in its subsequent development,i.e., we treat the droplet as a constrained thermodynamicquasi-equilibrium both internally or externally. However, theemergence of the cluster is actually a transient stochasticmechanism with a very short lifespan (typically a nano- oreven a picosecond) which depends essentially on the dropletradius. Apart from the fundamental query about the relevance

FIG. 6. Variation of non-dimensional length of triple line, L for a conicalcrest (a) and trough (b).

of using the conventional energetics to such formation, clearly,a more thorough understanding of the process of condensationnecessitates a kinetic model using which the growth rate ofthe cluster can be estimated due to the variation in theirsize. Moreover, the kinetic mechanism of condensation can betreated as a “Markov process” which can further be formulatedusing the Becker-Döring equations.2 In this connection, itcan be mentioned that the thermodynamic and the kineticaspects of condensation are linked through the exponentialdependence of the equilibrium cluster size distribution withthe free energy barrier which is a self-consistent assumptionwithin the framework of the thermodynamic fluctuationtheory.1

The following assumptions were espoused for the presentkinetic model: (1) nucleation of the incipient clusters canappear only at the favorable nucleating locations whichare finite in number, (2) the fundamental mechanism whichchanges the size of a cluster is the addition to it or detachmentfrom it by one monomer, (3) the mutual interactions amongthe clusters can be considered as negligible.

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204703-7 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

From the kinetics of heterogeneous condensation, asmentioned earlier, the attachment of monomers to the dropletcan occur by two means:22–25 (i) the explicit interfacialaccumulation on the neighborhood of the droplet-vaporinterface from the supersaturated vapor and (ii) the implicitperipheral addition via the vicinity of the triple line fromthe adsorbed layer of monomers on the substrate. Therefore,the attachment rate of monomers, also widely known as theforward rate or the condensation coefficient, can be given as22

a = asurf + aperi, (21)

where the contribution of the attachment rate based on theinterfacial accumulation of monomers on the neighborhoodof the droplet-vapor interface can be expressed as21

asurf = ανAαβ (r,ψ) . (22)

The contribution of the attachment rate due to theperipheral addition via the vicinity of the triple line canbe expressed as22

aperi = δν exp(∆Gads − ∆Gdiff

kBT

)L (r,ψ) , (23)

where the monomer impingement rate or the flux of monomersper unit interfacial area and per unit time can be given asν = pv√

2πmmkBT.24 For the nomenclature of the above terms,

one may refer Table III.From the study of energetics in Sec. III, it can be

clearly understood that ∆p, the difference in pressure betweenthe droplet and the supersaturated water vapor, is theprincipal driving thermodynamic force1 for condensation tooccur. This difference in pressure can also be expressed as∆p ≈ (kBT ln S) /Vm. Now, as the supersaturation, S = pv/psat

v ,the monomer impingement rate, ν = S ·psat

v√2πmmkBT

= S · νsat.

Therefore, monomer attachment rate at saturated ambientcondition can be evaluated as asat = asat

surf + asatperi. Finally,

monomer attachment rate can be rewritten as22

a = asat · S. (24)

In this context, a quantitative comparison between thetwo processes of droplet growth on a conical vertex can bemade, i.e.,

η =aperi

asurf=

(δ

α

)· exp

(∆Gads − ∆Gdiff

kBT

)·(

LAαβ

). (25)

From Eq. (25), η is found to have positive value and it is inthe order of 100 indicating that the rate of peripheral additionof the monomers has a much larger influence than the rateof interfacial accumulation of the monomers on the kineticsof the heterogeneous condensation. Moreover, the process ofthe peripheral accumulation becomes more dominant than theprocess of the interfacial addition when the size of the dropletis smaller. It can also be seen that η is inversely proportionalto the size of the cluster, which in turn indicates that theperipheral contribution to the monomer addition would behigher when the cluster is smaller.

To obtain the monomer detachment rate, usually classicaltheory of condensation employs the notion of constrainedequilibrium, despite the fact that the metastable supersaturated

vapor at a given temperature is not basically an equilibriumstate. In a different method derived by Katz withoutconsidering “Maxwell’s demon,” the detachment rate is foundfrom the “detailed balance” situation at the stable equilibriumstate of the saturated vapor at the same temperature.1 We haveused the Katz kinetic approach18 to determine the detachmentrate of monomers from the droplet in the present paper. So,monomer detachment rate for explicit interfacial accumulationcan be expressed as

dsurf = asatsurf · ω (26)

and monomer detachment rate for implicit peripheral additioncan be considered as

dperi = asatperi · ω, (27)

where ω is a kinetic factor obtained from the analysis ofdetailed balance.25

In this context, it can be mentioned that Luo et al.25

had considered Girshick-Chiu expression31 for the clustersize distribution in their kinetic model for heterogeneousnucleation on a spherical substrate. However, we have adoptedDillmann-Meier expression32 for the equilibrium numberdensity of clusters to determine the kinetic parameters. Itcan readily be shown that the kinetic factor (ω) becomesthe same if either of the expressions has been employed.Moreover, only the pre-exponential factor is different in boththe expressions. However, the factor is explicitly a functionof supersaturation and temperature. As the variation of thevarious kinetic parameters with respect to the line tension,and the substrate conicity and wettability are considered inthe present study, Dillmann-Meier approach is used in thepresent formulation for brevity.

For the present three-phase metastable system with theradius of the water droplet higher than 5 Å, the kinetic factorω can be defined as

ω = exp(

1kBT

· ∂∆G∂n

�����

sat

S=1

)= exp

(1

kBT· ∂∆FH

∂n

)= exp

(1

kBT· ∂∆FH

∂r· ∂r∂n

), (28)

where n is the number of monomers within the droplet.So, using Eqs. (B1), (B3), (B6), (B8), and (13), the kinetic

factor can be formulated as

ω = exp(

1kBT

· 8πσrf · Vm

4πr2

)= exp

Vm

kBT· 2σ

r

= exp(Ke), (29)

where Ke (= 2σVm/rkBT) is the Kelvin number which isequal to 0.7 in the present case.

Interestingly, the kinetic factor becomes the same forspherical—as well as conical-substrate-mediated dropletnucleation. Hence, by combining both the contribution tothe cluster growth, the detachment rate can be written as

d = dsurf + dperi, (30)

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204703-8 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

where the detachment rate associated with the direct interfacialaddition can be formulated as

dsurf = asatsurf · exp (Ke)

=psatv√

2πmmkBT· α · exp

(2σVm

rkBT

)· Aαβ (r,ψ) (31)

and monomer detachment rate due to the indirect peripheraldiffusion can be expressed as

dperi = asatperi · exp (Ke)

=psatv√

2πmmkBT· δ · exp

(∆Gads − ∆Gdiff

kBT

)· exp

(2σVm

rkBT

)· L (r,ψ) . (32)

Finally, monomer detachment rate, commonly knownas the backward rate or the evaporation coefficient, can beevaluated as

d = asat · ω = asat · exp (Ke) . (33)

Fig. 7 depicts the variation of the attachment anddetachment rate coefficients with respect to the radius of thedroplet, r and the microscopic wetting angle, θ for crest (a)and cavity (b) with different conicities and configurations. Thedetachment rate coefficient, d decays exponentially whereasthe attachment rate coefficient, a increases almost linearlywith respect to the size of the droplet, r . Besides, both therates decrease with the substrate hydrophobicity in case of aconical crest as can be seen from Fig. 7(a). On the other hand,the rates decrease with the substrate hydrophilicity duringheterogeneous condensation on a conical trough (Fig. 7(b)).As expected, both the attachment and the detachment ratesbecome identical at the critical radius for a given substratewettability.

B. Cluster growth rate

From the kinetics of the nucleation, it is inevitable thatthe formed cluster may grow or shrink depending upon itssize. So, the net growth rate of the droplet can be simply givenby the difference between the attachment and the detachmentrates for the cluster having radius typically higher than 5Å. The characteristics of the growth of a droplet dependessentially on the Knudsen number, Kn, which can be definedas the ratio of the molecular mean free path (λ) of watervapor molecules to the critical diameter (2rc) of the formeddroplet.25 Although the explicit expression for λ of watermolecules are unavailable in the literature, λ is considered asabout 42 nm from the kinetic theory of gases for the givensaturated vapor pressure (psat

v ). From Table I, the diameter ofthe droplet is found to be 3 nm. Therefore, the mean freepath is much larger than the molecular size of the dropletas Kn ≤ 14 during the period of the droplet growth. So,the growth is controlled by kinetics of the monomers in thesurrounding supersaturated vapor.33 Therefore, consideringthe present thermokinetic model as kinetically controlled orballistic limited growth, the growth rate of droplet can beapproximated to be proportional to the difference of themonomer attachment and detachment rates34 and can be

FIG. 7. The monomer attachment rate, a and the monomer detachmentrate, d as a function of the radius of the droplet, r and the microscopicwetting angle, θ. (a) Heterogeneous condensation on a conical vertex withvertex-angle 60◦, blue—case B, green—transition from case B to case C,red—case C and (b) heterogeneous condensation on a conical cavity withcavity-angle 60◦, red— case E, green— transition from case E to case F,blue— case F.

expressed using Eq. (13) as

rhet =

(drdt

)=

(∂r∂n

)·(

dndt

)=

Vm

4πr2 f· (a − d)

=Vm

4πr2 f· {S − exp (Ke)} asat. (34)

During the growth period,35 the influence of line tensioncannot be disregarded, as the droplet is within the nano tosub-micron level. It is true that that there is no explicit relationbetween the growth and the line tension as seen from Eq. (44).However, there is always a one-to-one relationship among r ,θ, and γ from Eq. (9) when φ, θ∞, and σ are kept as constants.Therefore, not only the growth rate but also the monomerattachment and detachment rates are implicitly related to

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204703-9 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

the line tension. For homogeneous condensation, the clustergrowth rate can readily be formulated from Eq. (44) as

rhomo = Vm · νsat · {S − exp (Ke)} . (35)

Fig. 8 shows the variation of the nanodroplet growth rateas a function of substrate wettability and conicity for bothconical crest and trough. The trends in the curves of Fig. 8are similar to that of Fig. 6 as the effect of triple line becomesmore important than the other geometrical parameters andthe mechanism of the peripheral accumulation becomes moredominant than the mechanism of the interfacial addition. Incase of the conical cavity, the growth rate suddenly increases inall the curves at the lower limiting condition due to associatingnon-activated barrierless condensation phenomena. Similarly,in case of the conical vertex, the growth rate almost vanishes inall the curves at the upper limiting condition due to associatedhomogeneous condensation phenomena. Besides, the growthrate increases with the substrate hydrophilicity in case ofcondensation on a conical crest (case B), although it increases

FIG. 8. Variation of growth rate of cluster with respect to wettability andconicity of different substrates: (a) crest and (b) trough.

with the substrate hydrophobicity in case of the condensationon a conical trough (case F) for cone-angle less than 90◦.On the other hand, though similar trends can be observedfor cone-angle greater than 90◦, there exists a maximum atθ = φ (case B) and θ = 180◦ − φ∗ (case F) for conical apexand pit, respectively. Moreover, an increment in the conicityenhances the growth rate considerably for a given substratewettability irrespective of the conical geometry. However, thegrowth rate increases monotonically with the hydrophilicityand the hydrophobicity for the cases C and E, respectively, asshown in Fig. 8.

C. Steady-state nucleation rate

Rate of nucleation (J) is one of the major determiningparameters for the kinetics of heterogeneous condensation.In case of homogeneous nucleation, the steady-stateheterogeneous nucleation rate is the number of nuclei formedper unit time per unit volume under the steady constrainedequilibrium condition of the three-phase metastable system.1

Nevertheless, the steady state nucleation rate can be definedas the number of nucleation events per unit area, per unit timeof the substrate for heterogeneous nucleation on a planarsubstrate.2 Similarly, nucleation rate can be evaluated interms of particle−1 s−1 in case of spherical-seed-mediatednucleation.4 However, the apices of the cones for crests aswell as troughs are distinct sites, i.e., not unvarying areas, forthe present configuration. Then the effect of the substrate areaincorporate via the nucleation site density (=NS)41 which canbe expressed as the number of the nucleation sites per unitprojected area, i.e., not the fractal42 area. As the nucleation sitedensity depends on the surface topography of the substrate,it is assumed to be 1014 sites m−2 for the present case.41 Theconsideration of this presumed value of the nucleation sitedensity is convenient because the present study focuses onthe effect of the various geometric parameters on conical-texture-mediated nucleation. So, the rate of nucleation can beexpressed in terms of site−1 s−1.

Within the framework of the classical theory ofheterogeneous nucleation, the steady-state heterogeneousnucleation rate can be expressed as25,36

J =Z · N (rc) · a (rc)

NS= J0 exp

(−∆Gc

kBT

), (36)

where the Zeldovich factor1 can be given by

Z =

− *,

12π· 1

kBT· ∂

2∆G∂n2

�����n=nc+-, (37)

the equilibrium cluster size distribution (Dillmann-Meierexpression) can be expressed as32

N = n2/3v exp

(−∆Gc (rc)

kBT

), (38)

the pre-exponential factor can be considered as2

J0 = n2/3v a (rc) Z/NS. (39)

Substituting value of ∂2∆G/∂n2 from Eq. (14) inEq. (37), the Zeldovich factor, based on classical theory

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204703-10 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

of heterogeneous nucleation, can be given by

Z == Zhom ·1

f, (40)

where the Zeldovich factor for homogeneous nucleation canbe evaluated as1

Zhom =

σ

kBT· Vm

2πrc2 , (41)

and nv is the number density of the supersaturated vapor andcan be expressed by assuming the metastable phase as idealgas as

nv = pv/kBT = Spsatv /kBT. (42)

Finally, the expression for steady-state heterogeneousnucleation rate can be evaluated explicitly as

J = αperiJ ′, (43)

where the nucleation rate in the absence of line tension isexpressed as

J ′ = J0 exp�−�∆Gc

hom/kBT�· f

�(44)

and the multiplying factor associated with the line tension isformulated as

αperi = exp (− (πrcγ/kBT) · g) . (45)

Fig. 9 depicts the variation of the steady-state heteroge-neous nucleation rate without considering the effect of linetension. From Eq. (44), it can be easily conceived that alower free energy barrier significantly enhances the rate ofnucleation regardless of the various configurations when linetension is absent. This can also be justified by comparingthe graphical plots shown in Figs. 4 and 9. It is seen fromFig. 9(a) that the rate has higher values for lower θ andhigher φ. J ′ is much higher for the conical cavities with lowersubstrate wettability and lower conicity without line tensioneffect as can be seen from Fig. 9(b). This also suggests that theprobability of nucleation to occur is higher within the troughsrather than the crests for textured substrates, i.e., conicalcavities are the most preferred locations for nucleation. Fora given substrate wettability, steady-state nucleation rate forplanar-substrate-mediated nucleation is always in betweenthat of the conical asperities and cavities irrespective of theconicities.

The influence of line tension on heterogeneous condensa-tion not only depends on the magnitude of the line tension butalso on its sign. It is fairly well-known that the magnitude ofline tension (γ) can be as small as 10−13–10−9 N irrespective ofits sign.37–40 Nevertheless, these values of line tension are notexplicitly considered for evaluating the peripheral contribution(αperi) to the nucleation rate in the present study, rather thevariation of αperi is shown in terms of the macroscopic contactangle (θ∞) using the modified Young’s equation, i.e., Eq. (3).

Figs. 10 and 11 illustrate the variation of the multiplyingfactor associated with the line tension for conical crest andtrough, respectively. From the modified Young’s equation,it can be seen that the two constants, i.e., γ and θ∞, haveone-to-one correspondence. Therefore, two different values ofθ∞ are considered in Figs. 10 and 11. The regimes of positive

FIG. 9. Illustrative depiction of the nucleation rate for crest (a) and trough(b) without considering the effect of line tension.

and negative line tension can be clearly seen in these figures.It can be discerned that the nucleation rate enhances and inturn promotes condensation for negative line tension (θ < θ∞)as it reduces the free energy barrier considerably. Similarly,condensation suppression can be possible for positive linetension (θ > θ∞) due to the increment in the free energybarrier. Interestingly, αperi becomes zero for cases C and Ebecause the effective contribution due to the line tension isabsent for these configurations as seen from Table II. So,Figs. 10 and 11 are applicable to the cases B and F only andthe corresponding values of the microscopic contact angle (θ∗)for the extremum value of the peripheral contribution can beobtained, for a given wettability and conicity, by solving thefollowing equations:

sin (2θ∗ − φ) − sin (θ∗ − φ) (2 cos θ∞ − cos θ∗)= 0, for case B (46a)

and

sin (2θ∗ + φ∗) − sin (θ∗ + φ∗) (2 cos θ∞ − cos θ∗)= 0, for case F. (46b)

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204703-11 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

FIG. 10. Illustration of the multiplying factor related to the line tension inEq. (45) for conical crest with θ∞= 60◦ (a) and 120◦ (b).

V. DISCUSSION ON PREFERRED NUCLEATION SITE

Fig. 12 depicts the various configurations rearrangedwithout the effect of line tension in such a way that thegeometric factor continuously decreases and in the wholerange of the appreciable configurations for a given value ofthe substrate conicity. So, the geometric factor, f varies from1 to 0 for promoting nucleation as shown in Fig. 12. Inthe present analysis, we have mainly shown the quantitativecharacteristics of nucleation is at and away from the apexof the cone for both the conical crest (cases B and C) andtrough (cases F and E). As mentioned earlier cases A, D,and G are the limiting cases of the present study. It is veryreasonable to exclude case A in the present formulation as thisconfiguration falls under the homogeneous nucleation regimeand this configuration is the most unfavorable configurationof all the possible cases.

Case B is the second most unfavorable configuration ascan be seen in Fig. 12. In this case, nucleation and subsequentgrowth of the cluster starts from the apex or tip of the conical

FIG. 11. Illustration of the multiplying factor related to the line tension inEq. (45) for conical trough with θ∞= 60◦ (a) and 120◦ (b).

microstructures. However, from the geometric model it isfound that for case C, the conical crest pierces the cluster,as can be seen from Fig. 12, suggesting that a droplet sittingon the apex of a cone is an unstable as well as unfavorableconfiguration (case B). Moreover, case C can be considered asa metastable state because if the cluster comes down over the

FIG. 12. Illustrative depiction of variation of controlling nucleation for dif-ferent configurations, but for a given conicity. These are the all possible sim-plified configurations for heterogeneous nucleation on conical (right circular)textures. Blue and red arrow indicate nucleation promotion and suppression,respectively.

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204703-12 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

slant height of the conical textures its change in the criticalfree energy diminishes. It can be easily discerned that aroundthe apex of a cone is the least likely site to nucleate ona patterned substrate without the effect of the line tension.When line tension is ignored, away from the apex of the conecan be considered as more potent nucleating spots for tworeasons: (i) the conical slant surface away from the tip ofthe cone is a more preferred location for nucleation events tooccur per unit area due to the reduction in the curvature and(ii) the conical slant surface away from the apex has higherarea as compared to that of its tip.

Line tension can play a vital role when it is considered inthe present thermokinetic model. Higher value of positive linetension stabilizes the formed droplet at apex (case B), whereasit destabilizes the configuration corresponding to case C ascan be seen from Figs. 3(a) and 3(b). Similarly, higher valueof negative line tension favors considerably the formation andevolution of the cluster related to case C and a transitionfrom case B to case C is inevitable. The thermokineticcharacteristics of the planar-substrate-induced nucleation(case D), which is a classical limiting case of the presentformulation, is always in between that of heterogeneousnucleation on the conical vertex and cavity. Interestingly, theequilibrium condition or the modified Young’s equation forcase D becomes the same for cases C and E. Moreover, planarsurface is always more preferred nucleation location than theconical asperity, while it is more unfavorable nucleation spotthan the conical cavity.

It can be easily perceived from Fig. 12 that conical cavityacts as better preferred site than the conical asperity. Recently,a three stage mechanism is described by Yarom and Marmurfor a hydrophilic conical pore or cavity.43 In this scenario,the configurations corresponding to cases E and F can beconsidered as Cassie and Wenzel state, respectively.44 So,similar to case C for conical asperity, case E can be consideredas a metastable state. However, case F for cavity is stable statewhile case B for asperity is unstable state as discussed earlier.As similar as nucleation on the conical asperity, line tensionhas considerable effect for conical-cavity-induced nucleation.Higher value of positive line tension stabilizes the formeddroplet away from the apex (case E), whereas it destabilizesthe configuration corresponding to case F as can be seen fromFigs. 3(d) and 3(c). Similarly, higher value of negative linetension prefers the formation and the subsequent growth ofthe cluster related to case F and a transformation from case Eto case F can be expected. So, hydrophilic narrower conicalcavity is the most preferred site for nucleation to occur andthe negative line tension acts as an extra driving force fornucleation.

The configuration for case G is a limiting condition of thepresent thermokinetic model and widely known as capillarycondensation, barrierless nucleation, athermal nucleation,non-activated nucleation, etc. This is the situation for whichthe interface of the nucleating droplet within a conical troughbecomes concave. Turnbull19 studied the energetics of acrystalline nucleus emerging from its own melt, but the sameaspects hold for droplet nucleation from its oversaturatedvapor. Stabilized by the negative supersaturation near thespinodal limit, nanodroplets with a concave interface existing

at the bottom of a conical trough are more stable than thehomogeneous phase. The embryos are seemed to surviveunder situations when they might have been supposed to beshrunk. This is obviously crucial for potential understandingof the “memory effects” on nucleating substrates.45–48

VI. CONCLUSIONS

In the present paper, we have discussed a novelthermokinetic model for heterogeneous droplet nucleation onconical textures for different geometric configurations withinthe framework of classical theory of heterogeneous nucleation.Both the processes of monomer exchange along with theinfluence of peripheral tension have been considered in theanalytical derivation. A rigorous and systematic energeticformulation based on the principle of the extremization of thefree energy has been developed to relate the microscopicwetting angle and the line tension to find the criticalparameters of nucleation. Further, a kinetic model based onthe notion of the detailed balance of monomer exchange hasalso been formulated to determine the various associatedkinetic rate coefficients, i.e., monomer detachment rate,cluster growth rate, and steady-state nucleation rate. We haveused two nondimensional numbers, i.e., Kelvin number (Ke)and Knudsen number (Kn), to characterize the kinetics ofnucleation. The following points can be summarized from ourthermokinetic model:

(1) Irrespective of configurations, the free energy barrier haslower values and the nucleation rate has higher valuesfor lower substrate wettability and higher conicity forcrest and lower conicity for trough in the absence ofline tension. There is a considerable effect of triple lineon both thermodynamics (as line tension) and kinetics(as peripheral exchange of monomers) of conical-texture-induced nucleation.

(2) The implicit peripheral exchange of monomers via thevicinity of the triple line from the adsorbed layer isfound to be almost two orders of magnitude higherthan the implicit interfacial exchange at the droplet-vaporinterface. The peripheral component of the free energybarrier and the droplet growth rate is proportional to thelength of the triple line.

(3) The substrate hydrophobicity and hydrophilicity are foundto switch over their roles between conical crest and troughfor different growth rates of the cluster. For a negativeperipheral tension, the rate of nucleation enhances andin turn promotes nucleation as it reduces the free energybarrier considerably, whereas nucleation inhibition maybe possible for positive peripheral tension values due tothe increment in the free energy barrier.

(4) For a given wettability and conicity, the correspondingvalues of the microscopic contact angle (θ∗) for theextremum value of the peripheral factor to the nucleationrate can be obtained from extremization principle.Analyzing all possible geometric configurations, thehydrophilic narrower conical cavity is found to be themost preferred site for nucleation to occur.

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204703-13 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

Therefore, the present thermokinetic model can be usedas a guideline for the design of engineered substrates topromote or to suppress droplet nucleation. Moreover, thepresent formulation can be further used to study the variationof nucleation rate for different values of peripheral tensionand work in this direction is in progress.

ACKNOWLEDGMENTS

We like to appreciate the anonymous referee forthoughtful comments and suggestions for the improvementof the manuscript.

APPENDIX A: DERIVATION MODIFIED DUPRE-YOUNGEQUATION

From the principle of the Lagrange multiplier technique,the conditions can be written as

∂E∂r= 0 and

∂E∂θ= 0, (A1)

where E = ∆FH − λV and λ is the Lagrange multiplier which,in turn can be shown that, is nothing but ∆p, the differencein pressure between the stable cluster and the supersaturatedwater vapor.

So, E can be rewritten for case B as

E (r, θ) = σ

2πr2 (1 + sin (θ − φ))

−(cos θ∞ × πr2 · cos2 (θ − φ)

sin φ

)+ (γ × 2πr cos (θ − φ))

−λ × π3

r3 ·(2 + 3 sin (θ − φ) − sin3 (θ − φ)

− cos3 (θ − φ) · cos φsin φ

). (A2)

Therefore, ∂E∂r= 0 gives

λ =2σr

2 (1 + sin (θ − φ)) − cos θ∞ · cos2(θ−φ)

sinφ

+ 2γ cos (θ − φ)

r2(2 + 3 sin (θ − φ) − sin3 (θ − φ) − cos3 (θ − φ) · cosφ

sinφ

) (A3)

and ∂E∂θ= 0 gives

λ =2σr cos (θ − φ) 1 + cos θ∞ · sin(θ−φ)

sinφ

− 2γ sin (θ − φ)

r2cos2 (θ − φ) (cos (θ − φ) − sin (θ − φ) · cosφsinφ

) .

(A4)

Equating both the λs from Eqs. (A3) and (A4) and aftersome extensive algebraic exercise, Eq. (8) can be derived.Similarly, modified Young’s equation can be derived for caseF by simply substituting 180◦ − φ∗ in place of φ.

Furthermore, E can be reformulated for case C as

E (r, θ) = σ �4πr2 sin θ cos φ −

�cos θ∞ × 4πr2

· cos θ sin θ cos φ ) } + (γ × 4πr cos θ cos φ)−λ × 4π

3r3 · sin3θ cos φ. (A5)

So, ∂E∂r= 0 gives

λ =2σr sin θ (1 − cos θ∞ cos θ) + γ cos θ

r2sin3θ(A6)

and ∂E∂θ= 0 gives

λ =σr2 �cos θ − cos θ∞

�cos2θ − sin2θ

��− γ sin θ

r3sin2θ cos θ. (A7)

Equating both the λs from Eqs. (A6) and (A7) and aftersome extensive algebraic exercise, Eq. (8) can be derived.Within the same principle, the modified Young’s equation canbe derived for case E.

APPENDIX B: EVALUATION OF VARIOUS PARTIALDERIVATIVES OF HELMHOLTZ AND VOLUMETRICFREE ENERGIES WITH RESPECT TO THE RADIUSOF THE CLUSTER

The partial derivatives of the free energy contributionswith respect to the radius of the cluster for case B can beevaluated as∂∆FH

∂r= σ

4πr (1 + sin (θ − φ)) −

(cos θ∞

× 2πr · cos2 (θ − φ)sin φ

)+ (γ × 2π cos (θ − φ))

= 2πσr ×

2 (1 + sin (θ − φ))

− cos θ∞ ·cos2 (θ − φ)

sin φ

+ 2πσr · cos2 (θ − φ)

sin φ{cos θ∞ − cos θ}

= 8πσr ·

14

(2 + 3 sin (θ − φ) − sin3 (θ − φ)

−cos3 (θ − φ) · cos φsin φ

)= 8πσr · f (θ,φ) (B1)

and∂∆FV

∂r= 4π∆pr2 ·

14

(2 + 3 sin (θ − φ) − sin3 (θ − φ)

−cos3 (θ − φ) · cos φsin φ

)= 4π∆pr2 · f (θ,φ) . (B2)

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204703-14 Singha, Das, and Maiti J. Chem. Phys. 143, 204703 (2015)

Similarly, the partial derivatives corresponding to case Fcan be obtained by simply substituting 180◦ − φ∗ in place ofφ and those can be expressed as

∂∆FH

∂r= 8πσr · f (θ,φ∗) (B3)

and

∂∆FV

∂r= 4π∆pr2 · f (θ,φ∗) . (B4)

Moreover, substituting the value of line tension fromthe modified Dupre-Young equation in the expression of theHelmholtz free energy for case C and after some algebraicmanipulation one can obtain

∆FH = 4πσr2 · sin3θ cos φ. (B5)

Therefore, the partial derivatives of the free energycontributions with respect to the radius of the cluster forcase C can be evaluated as

∂∆FH

∂r= 8πσr · sin3θ cos φ = 8πσr · f (θ,φ) (B6)

and

∂∆FV

∂r= 4π∆pr2 · sin3θ cos φ = 4π∆pr2 · f (θ,φ) . (B7)

Similarly, the partial derivatives corresponding to case Dcan be obtained as

∂∆FH

∂r= 8πσr · f (θ,φ∗) (B8)

and

∂∆FV

∂r= 4π∆pr2 · f (θ,φ∗) . (B9)

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