ARTICLE pubs.acs.org/JPCB
Thermodynamic Investigation of the Barrier for Heterogeneous Nucleation on a Fluid Surface in Comparison with a Rigid Surface Fatemeh Eslami and Janet A. W. Elliott* Department of Chemical and Materials Engineering, University of Alberta, Edmonton AB, Canada T6G 2V4 ABSTRACT: When a vapor phase is in contact with a solid or nonvolatile fluid, under conditions where the vapor is thermodynamically metastable to condensation, a droplet may nucleate from the vapor either homogenously within the vapor phase, or heterogeneously at the solid or fluid substrate interface. The case where the droplet is thermodynamically favored to nucleate heterogeneously is the subject of this article. The heterogeneous nucleation of a sessile drop on a soft surface has been studied many times experimentally and theoretically. It has been observed experimentally that heterogeneous nucleation happens faster on a soft surface in comparison with a rigid surface. Here we use Gibbsian surface thermodynamics to provide a physical understanding for this observation. Due to the difficulties of considering soft-elastic surfaces, we demonstrate that by considering only the fluidity of a surface (i.e., by considering a fluid surface as an infinitely soft material and comparing a fluid surface with a rigid surface), thermodynamics will predict that heterogeneous nucleation is easier on soft surfaces compared with rigid surfaces. We first investigate the effect of contact angle on the barrier for heterogeneous nucleation on rigid substrates at constant vapor phase pressure. Then we find a lower energy barrier for heterogeneous nucleation at a fluid surface in comparison with heterogeneous nucleation at a rigid surface which explains the faster nucleation on soft surfaces compared with rigid surfaces. Finally we inspect the role of each contribution to the energy barrier.
’ INTRODUCTION Nucleation is one of the processes accompanying first-order phase transitions.1 Formation of a bubble in a liquid phase, or a drop in a vapor phase, is an example of homogeneous nucleation. Heterogeneous nucleation, the topic of interest in this article, takes place in the case of the presence of a third phase (usually solid), which provides sites for nucleation. Heterogeneous nucleation can be important2 in heat transfer3 and vapor deposition.4 Nucleation on soft surfaces is important since many natural and technological substances such as tissues and polymers are soft materials. In addition, at very small scales such as nanometers, many of the substances that seem to be rigid act like soft matter, as interfacial forces overcome the strength of the small amount of material. Due to the importance of nucleation of a sessile drop on soft surfaces, several works have been done in this area. However, most researchers treat the soft surface as an elastic solid. The sessile drop deforms a soft surface, the deformation being the result of the vertical component of surface tension around the contact line, which pulls the solid upward, and also the Laplace pressure inside the drop, which pushes the solid downward.5 Several mathematical functions describing the surface deformation have been proposed.69 There are two approaches to find the equilibrium shape of a soft surface on which a liquid droplet rests. One approach is a mechanical approach via force balances and the other one is a thermodynamic approach through minimization of free energy. Shanahan10 is one of the researchers r 2011 American Chemical Society
that applied the minimization of free energy to explore the nucleation of a drop on a soft surface. He considered interfacial free energy and the elastic energy of deformation in the free energy term and, by imposing constant volume as a constraint and applying the calculus of variations, minimized the free energy. He found a modification for the Young equation for the mesoscopic scale and attributed the contact angle change with drop size to the role of Laplace pressure at the mesoscopic scale rather than line tension. Lester6 and Rusanov7 considered a semi-infinite surface for nucleation of a drop and by means of stress analysis found the deformation profile for the surface. However, in order to avoid infinite stress at the triple line, they had to consider a thickness (t) for the liquidvapor interface. Lester concluded that the Young equation is not applicable unless deformation of the surface does not happen. On the other hand, Shanahan and de Gennes11 performed stress analysis and found a logarithmic deformation profile. They pointed out that in the vicinity of the triple line, linear elasticity is invalid since the radial stress will go to infinity. Therefore, they considered a small distance (ε) around the triple line in which linear elasticity is not applicable, and instead plastic or other nonlinear elasticity governs that area. Alternatively, White12 Received: March 2, 2011 Revised: June 5, 2011 Published: July 07, 2011 10646
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The Journal of Physical Chemistry B believes that the reason for which these unclear length scales such as t and ε are brought into the calculations is the absence of intersurface forces at the microscopic scale. He considered the transmission force of the surface tension to the substrate by introducing the effect of disjoining pressure to his calculations and proposed profiles for the drop and substrate in integral forms. In addition to semi-infinite soft surfaces, some researchers investigated the deformation of thin elastic surfaces13 such as thin plates14 and membranes8 under the presence of a sessile drop. It was found that the deformation profile for the sessile drop on the thin plates has a pseudoparabolic9 shape, but for the membranes the ridge will still appear around the triple line. Yu and Zhao8 studied the effect of the thickness of the membrane on its deformation shape. They figured out that each membrane has a “saturated thickness”. If its thickness is thicker than that, it can be considered as a semi-infinite body; otherwise thickness has an important effect on the deformation of the solid induced by a water droplet. Besides these various deformation profiles for substrates, there exist controversial discussions about the contact angle of sessile drops on soft surfaces. In addition to these theoretical works on the nucleation of a drop on a soft surface, several experimental investigations have been done. For example the presence of the ridge around the triple line was observed by means of scanning white light interferometric microscopy by Carre et al.15 They found a good conformity between their observations and their estimation for its height. Based on their estimation, ridge height is proportional to the vertical component of surface tension and inversely proportional to the shear modulus of the substrate (G). Sokuler et al.2 performed experiments and recognized that nucleation happens more easily on soft surfaces compared to rigid ones. They manufactured a soft surface and observed that on the softer surfaces the nucleation density is higher. They suggested that this phenomenon can be attributed to a lower barrier for nucleation, which originates from the lower free energy in the case of soft surfaces. They mention that, although for soft surfaces the system has an extra energy of elasticity and the interfacial energy of the liquidsolid interface is larger due to its higher area, the smaller liquidvapor interfacial area due to the presence of the ridge overcompensates these energies, and the system faces a lower free energy and hence the lower barrier. They also concluded that, for softer surfaces, coalescence of nucleates is delayed. Since the ridge height is inversely proportional to G, the higher the ridge, the later the nucleates merge, hence liquid covers a larger area on softer solids. On the whole, Sokuler et al.2 performed experiments and submitted a physical argument without any kind of mathematical derivation that nucleation happens easier on soft surfaces. Therefore, one of the goals of this paper is to prove this phenomenon theoretically by means of detailed surface thermodynamics. As we have discussed, the elasticity of the solid introduces several complexities in calculations. For example, some of the consequences of the elastic solid are (i) no explicit solid deformation profile, (ii) care should be taken in how contact angle is defined, and (iii) the presence of unclear length scales such as capillary layer thickness in the calculations. Here we plan to answer the question of whether nucleation is easier on a rigid surface or a soft surface. However, due to the difficulties of considering soft-elastic surfaces, we demonstrate that by considering only the fluidity of a surface (i.e., by considering a fluid surface as an infinitely soft material and comparing a fluid surface with a
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rigid surface), thermodynamics will predict that nucleation is easier on soft surfaces. Hence, we investigate the thermodynamic stability of a sessile drop on either a rigid surface or a fluid surface at constant vapor phase pressure and compare their energy barriers. We note that several works have been done before on homogeneous and heterogeneous nucleation. Of particular relevance, Nepomnyashchy et al.16 compared the free energy barrier of homogeneous nucleation and heterogeneous nucleation at a liquid gas interface and showed that heterogeneous nucleation of a drop at a liquidgas interface is more favorable than homogeneous nucleation of a drop in the vapor phase; Forest17 and Ward investigated the nucleation of a bubble in liquidgas solutions both homogeneously and at the liquidliquid interface, showing that the surface and interfacial tensions must meet a specific criterion for the bubble to form heterogeneously; and Jarvis et al.18 showed that a liquid lens can exist at the interface if specific criteria are met for the magnitudes of surface and interfacial tensions (see Appendix for details). However, to the best of our knowledge, the comparison of heterogeneous nucleation on a rigid surface with that on a fluid surface has not been done previously. In this regard, we have chosen the surface tensions and interfacial tension such that the criteria for heterogeneous nucleation are met (see the Appendix for detailed discussion on the conditions imposed on surface and interfacial tensions), and we assume that either the liquid lens or the sessile drop forms at the interface rather than homogeneously so that we can compare the heterogeneous nucleation on the rigid surface with that on the fluid surface. There has been other research performed in the field of surface thermodynamic stability analysis of systems with curved surfaces. However, most of these works pertain to systems with finite size1923 or capillary bridges systems.24,25 Depending on the constraints and the conditions of the system, different free energies and stabilities are found for these systems.
’ GOVERNING EQUATIONS In order to determine the stability of equilibrium states, the second derivative of free energy with respect to size of the drop or bubble should be explored at the equilibrium radii, Rc. If the second derivative is negative, Rc represents an unstable equilibrium. If the second derivative is positive, Rc represents a stable equilibrium. Metastable equilibria are located at local minima in the free energy function. As a result, if a drop forms spontaneously in a vapor phase with a radius larger than the unstable size, it will grow larger. On the other hand, if a drop forms spontaneously with a radius smaller than the unstable size, the drop will evaporate. But if the drop radius is in stable equilibrium, the drop is unaffected by small perturbations (larger or smaller), after which it will return to the stable size. Generally, to perform the stability analysis the following procedure is followed: (i) derivation of conditions for equilibrium, (ii) combining conditions for equilibrium with equations of state to find the equation for the equilibrium radius, (iii) determining what function acts as the thermodynamic potential (or free energy), and (iv) use of free energy to determine the number and stability of equilibrium states. Herein, we perform these procedures first for the nucleation of a sessile drop on a rigid surface and then repeat it for nucleation on a fluid surface. Throughout, we neglect the effects of gravity due to the small size of the drops. 10647
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Nucleation of a Sessile Drop on a Rigid Surface. Figure 1a shows a sessile drop with radius R and contact angle θ on a rigid solid. The liquid phase and the vapor phase consist of component 1 and the solid phase consists of component 2. The surroundings, or “reservoir”, including the container, are considered to be another phase with constant pressure and temperature (PR and TR as shown in Figure 1). The reservoir can exchange energy and volume with the system through the walls of the system and movable piston. The system is closed and cannot exchange mass with the reservoir. The nature of interactions between system and reservoir dictate the constraints for the composite system. We know that at equilibrium, the entropy (S) of the system plus the reservoir together will be an extremum, so we require
(iv) The solid is nonvolatile and incompressible: dN2S ¼ 0
Z V ¼ L
r
∑ μij dNji j¼1
dU ab ¼ T ab dSab þ γab dAab þ
dU ab ¼ T ab dSab þ γab dAab þ
r
ab μab ∑ j dNj j¼2 r
∑ μabj dNjab
j¼1
ð2Þ
ð3Þ
ð4Þ
where superscript i denotes each bulk phase, i.e., liquid, vapor or solid, Nj is the number of moles of component j in phase i, γ is the surface tension, and the superscript “ab” identifies the interface subsystems, i.e., SL, LV, or SV. The constraints that are imposed on the system are described below. (i) The system and reservoir together are isolated: dU V ¼ dU L dU S dU LV dU SL dU SV dU R ð5Þ (ii) The system is closed, so that there is no mass exchange between the system and reservoir: dN1V ¼ dN1L dN1LV dN1SL dN1SV
ð6Þ
dN1R ¼ 0 dN2R ¼ 0 ð7Þ (iii) A moveable piston separates the system from the reservoir: dV V ¼ dV L dV S dV R
ð8Þ
θ
πR 3 sin3 j dj ¼
0
Z
ð1Þ
dU i ¼ T i dSi Pi dV i þ
ð9Þ
In addition, based on the “Gibbs dividing surface” approach, the excess surface quantity for N2 is taken to be zero for the flat SV interfaces SV and SL; therefore NSL 2 = 0 and N2 = 0. As a consequence of the existence of derivatives of bulk volumes and interfacial areas in relations 24, the geometry of the system must be specified. The drop volume and areas for the LV and SL interfaces are given by the following relations:22
dSL þ dSV þ dSS þ dSLV þ dSSL þ dSSV þ dSR ¼ 0 In eq 1, superscripts R, L, V, S, SL, LV, and SV represent the reservoir, liquid, vapor, solid, solidliquid, liquidvapor, and solidvapor phases, respectively. For a bulk phase, a flat interface, or a curved interface, the differential form of the fundamental equation of thermodynamics is given by eqs 2, 3, or 4, respectively, which relate changes in the volume, V, the area, A, and the thermodynamic properties temperature, T, pressure, P, and chemical potential, μ, to changes in the internal energy, U.
dV S ¼ 0
θ
ALV ¼
πR 3 ½2 cos θð2 þ sin2 θÞ 3 ð10Þ
2πR 2 sin j dj ¼ 2πR 2 ð1 cos θÞ
ð11Þ
0
ASL ¼ πR 2 sin2 θ
ð12Þ
In order to find the derivatives, the volume and areas should be differentiated with respect to both θ and R. Substituting equations of form 2 for the bulk phases (liquid, vapor, solid, and reservoir), equations of form 3 for the flat interfaces SV and SL, an equation of form 4 for the curved LV interface, the constraints 59 and derivatives of the geometry relations 1012 into eq 1 and collecting like terms we find: ! ! R V L V 1 1 P P μ μ 1 1 dN1L dV R dU L þ TL TV TR TV TL TV 1 1 1 1 S þ dU þ dU LV TS TV T LV T V ! μLV μV1 1 1 LV 1 þ dU SL dN 1 T SL T V T LV T V ! μSL μV1 1 1 SL 1 dN1 þ dU SV T SV T V T SL T V ! μSV μV1 1 1 SV 1 þ dU R dN 1 TR TV T SV T V " ! PL PV þ ð2 cos θð2 þ sin2 θÞÞπR 2 TL TV γLV ðγSV γSL Þ 2 ð2πR sin θÞ dR LV ð1 cos θÞ4πR þ T SL T ! " PL PV πR 3 V ðsin θð2 þ sin2 θÞ 2 sin θ cos2 θÞ þ L T T 3 # γLV ðγSV γSL Þ LV 2πR 2 sin θ þ ð2πR 2 sin θ cos θÞ dθ ¼ 0 T SL T
ð13Þ In order for the above expression to be valid for any arbitrary displacement about equilibrium, the coefficient of each independent variation may be set equal to zero. Consequently, the 10648
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After performing the calculations28,29 that determine which function acts as the free energy for this system, the following free energy,25,29 B, is found: ΔB ¼ ΔðGV þ F L þ F S þ F SL þ F LV þ F SV þ PV V L Þ e 0
ð20Þ
where G is the Gibbs free energy and F is the Helmholtz free energy. Since the energy should be computed with respect to a reference point, for convenience we choose the system of the solid and vapor phase without any drop in it to be a reference state; thus the potential function for the reference state is equal to V SV SV SV S S S S þ μSV B0 ¼ μV10 N10 10 N 10 þ γ A0 þ μ20 N20 P V
ð21Þ Given that the total moles and container cross sectional area will not change from that of the reference state, and by assuming the reference state to be an equilibrium state, we will have eq
eq
BðRÞ B0 ¼ N1V ðμV1 μ1 Þ þ N1L ðμL1 μ1 Þ eq
eq
SV SV þ N1LV ðμLV 1 μ1 Þ þ N1 ðμ1 μ1 Þ eq þ N1SL ðμSL 1 μ1 Þ SL SV SL
þ ðγ γ ÞA þ ðPV PL ÞV L
þ
eq N2S ðμS2 μ2 Þ LV LV
þ γ A
ð22Þ
Since equilibrium states are going to be investigated, the value of B(R) B0 in the limit of equilibrium is equal to25 BðRÞ B0 ¼ ðγSL γSV ÞASL þ γLV ALV þ ðPV PL ÞV L
Figure 1. Geometry for (a) nucleation on a rigid surface, and (b) nucleation on a fluid surface.
conditions for equilibrium are found to be as follows: T L ¼ T LV ¼ T SL ¼ T V ¼ T SV ¼ T S ¼ T R
ð14Þ
SV L SL μV1 ¼ μLV 1 ¼ μ1 ¼ μ1 ¼ μ1
ð15Þ
PV ¼ PR
ð16Þ
PL PV ¼
2γLV Rc
ð17Þ
γSV γSL ¼ γLV cos θ
ð18Þ
Equations 17 and 18 are the Laplace equation and Young equation, respectively, which come out of the derivation as conditions for equilibrium. After combining conditions for equilibrium with equations of state (ideal gas for the vapor, and incompressible fluid for the liquid), the following relation for equilibrium radii will be found:26 Rc ¼ "
2γLV !
PV RT ̅ ln νL∞ P∞
þ P∞ P
#
ð19Þ
V
In eq 19, P∞, νL∞ and R represent the saturation pressure, liquid molar volume at the saturation pressure, and the universal gas constant, respectively. Equation 19 is equivalent to what is commonly referred to as the Kelvin equation, which differs from the original Kelvin equation since it was corrected by Gibbs to be thermodynamically consistent.27,28
ð23Þ We note that the above approach implicitly assumes that the values of the intensive parameters at equilibrium are the same as those of the hypothetical reference equilibrium state. Such a reference state need not physically exist. Nucleation of a Sessile Drop on a Fluid Surface. Figure 1b shows a sessile drop of one liquid, L, on another liquid, L0 , with contact angle β and deformation angle R surrounded by a reservoir at constant pressure, PR, and constant temperature, TR. Here, the L and V phases consist of component 1, and the L0 phase consists of component 2. We repeat the same procedure as for the nucleation on a rigid surface. The constraints are the same as for the rigid surface except for that, in the nucleation on a fluid surface, only the L0 V interface is flat, so based on the “Gibbs dividing surface” approach 0
N2L V ¼ 0
ð24Þ
Therefore, the conservation of moles for component 2 becomes 0
dN2L ¼ dN2LL
0
ð25Þ 0
In addition, in this case, the constraint dVL = 0 is not imposed. The drop volume and interfacial areas in this case are given by the relations for liquid lenses:16 Z β Z R L 3 3 V ¼ πRU sin j dj þ πRL3 sin3 j dj 0
0
πRU3 ½2 cos βð2 þ sin2 βÞ ¼ 3 πR 3 þ L ½2 cos Rð2 þ sin2 RÞ 3 10649
ð26Þ
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β
ALV ¼ 0 0
Z
R
ALL ¼ 0 0
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2πRU2 sin j dj ¼ 2πRU2 ð1 cos βÞ
ð27Þ
2πRL2 sin j dj ¼ 2πRL2 ð1 cos RÞ
ð28Þ
0
0
ð29Þ
AL V ¼ AL0 V πRU2 sin2 β ¼ AL0 V πRL2 sin2 R 0
In eq 29, AL0 V is the initial L0 V interfacial area, before the liquid lens has formed. In addition, the following equation relates the geometrical variables: RU sin β ¼ RL sin R
ð30Þ
where RU is the radius of curvature of the upper curved interface, LV, and RL is the radius of curvature of the lower curved interface, LL0 . In order to find the derivatives of volume and areas, they should be differentiated with respect to three independent variables, which are any three variables out of RU, RL, R, and β. Following a similar procedure to that used for nucleation on a rigid surface, the equilibrium conditions for a sessile drop on a fluid interface are found to be 0
0
0
T L ¼ T LV ¼ T LL ¼ T V ¼ T L V ¼ T L ¼ T R 0
LV μV1 ¼ μLV ¼ μL1 ¼ μLL 1 1 ¼ μ1 0
μL2 ¼ μLL 2
ð31Þ
0
ð32Þ
0
ð33Þ
PV ¼ PR
ð34Þ
0
ð35Þ
PV ¼ PL
0
ðPL PV Þ ¼ ðPL PL Þ ¼
ðPL PV Þ ¼
Figure 2. Free energy B(R) B0 versus the radius of the water drop on a rigid surface for different contact angles where T = 22 °C, PV = 2700 Pa, and γLV = γw = 72.43 mN/m.
0
2γLL 2γLL ¼ sin β RLc RUc sin R
2γLV RUc
0
ð36Þ
0
0
ð38Þ
0
ð39Þ
γLV sin β ¼ γLL sin R
Equations 36 and 37 are the Laplace equations for the LL0 and LV interfaces, respectively, and eqs 38 and 39 are the force balance of interfacial tensions at the three phase contact line, in the horizontal and vertical directions; so they can be called Young equations. RUc and RLc in eqs 36 and 37 are the equilibrium radii of upper curvature and lower curvature, respectively. The same free energy B is found for the fluid case as in the rigid case, but due to the differences in geometry, B(R) B0 is found to be16 0
0
’ RESULTS AND DISCUSSION Effect of Contact Angle on the Energy Barrier for Heterogeneous Nucleation on a Rigid Surface. First, we investigate
ð37Þ
γLV cos β þ γLL cos R ¼ γL V
Figure 3. Free energy B(R) B0 versus radius of the water drop for fluid (dodecane) substrate and rigid substrate with the same interfacial 0 tensions where T = 22 °C, P0 V = 2700 Pa, γLV = γw= 72.43 mN/m, γL V = d LL wd γ = 25.3 mN/m, and γ = γ = 53.7 mN/m.
0
BðRÞ B0 ¼ γLL ALL þ γLV ALV γL V πRU2 sin2 β þ ðPV PL ÞV L
ð40Þ
the effect of contact angle of a drop on a rigid surface. We consider drops of water that are nucleated from their vapor at constant vapor phase pressure heterogeneously and on rigid solids on which they experience different equilibrium contact angles. By specifying the contact angles, we are considering different surfaces with different inherent surface tensions and adsorptive states that correspond to the chosen contact angle at the equilibrium state. Figure 2 shows the potential function B(R) B0 versus the drop radius R for different contact angles for a drop of water on a rigid surface at T = 22 °C and PV = 2700 Pa. The obtained equilibrium radius and liquid pressure are Rc = 4.9072 108 m and PL = 2.9547 106 Pa. From the extrema in Figure 2, an unstable equilibrium state is seen in the system. Thus, if a spontaneously formed drop has a radius smaller than the unstable equilibrium radius (the maxima in Figure 2) it will evaporate, but if its radius is larger than the equilibrium radius, the drop will grow larger, and since there is not any stable drop size in this 10650
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The Journal of Physical Chemistry B case, the drop will grow until all the vapor has condensed. Figure 2 shows that as contact angle is increased, drop nucleation becomes harder since the barrier for nucleation becomes larger. Comparison of the Barrier for Heterogeneous Nucleation on a Rigid Surface with That for a Fluid Surface. In order to compare the energy barrier for heterogeneous nucleation on a rigid surface with that for a fluid surface, we choose two systems. In one case, we have a container of dodecane, and a drop of water from its vapor is nucleated on it at constant vapor phase pressure (Figure 1b). In the second case, we have a rigid solid with the same solidliquid interfacial tension as dodecaneliquid water and the same solidvapor surface tension as dodecanewater vapor, and then we nucleate a drop of water from its vapor phase on it (Figure 1a). Therefore, by supposing fluids L and L0 to be water and dodecane, at0 T = 22 °C the surface tension of dodecanewater 0 vapor,30 γL V = γd, and waterdodecane,30 γLL = γwd, can be taken to be 25.3 mN/m and 53.7 mN/m, respectively.
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Other properties are taken to be γLV = γw = 72.43 mN/m and PV = 2700 Pa. Since we have specified the interfacial tensions of this hypothetical case, and we, by definition, consider these to be the surface tensions at the equilibrium state, the contact angle β and the deformation angle R that corresponds to these interfacial tensions follow from eqs 38 and 39 in the fluid case, and the contact angle, θ, follows from eq 18 in the rigid case. The contact angle (θ) in the rigid case is θ = 113°, while the contact angle (β) and the deformation angle (R) for the fluid surface are found to be β = 34.99° and R = 129.33°. Figure 3 shows the plot of B(R) B0 versus R; it confirms that the free energy extremum in the fluid surface case is at the same critical radius as for the rigid surface, but due to the lower barrier for the fluid surface, nucleation is predicted to happen more easily. In addition, a plot of B(R) B0 versus volume of the liquid phase (L) is given in Figure 4. It also demonstrates the ease of nucleation on a fluid surface in two ways. As can be seen, at equilibrium for the fluid surface case, not only is the barrier lower than for the rigid surface, but also the unstable equilibrium happens at a smaller drop volume. On the basis of Figures 3 and 4, an explanation is provided for the experimental work of Sokuler et al.2 which showed that nucleation is easier on soft surfaces. In order to further understand physically the reason for the lowered barrier for nucleation on fluid surfaces, we investigate the contribution of each term to the free energies in the rigid and fluid cases and plot each of these terms versus the radius of the drop for both cases. We name each of the terms as the following: For the rigid surface:
Figure 4. Free energy B(R) B0 versus water drop volume for fluid (dodecane) substrate and rigid substrate with the same interfacial tensions where T = 22 °C, PV = 0 2700 Pa, γLV = γw = 72.43 mN/m, 0 γL V = γd = 25.3 mN/m, and γLL = γwd = 53.7 mN/m.
BLaplace ¼ ðPV PL ÞV L
ð41Þ
BLV ¼ γLV ALV
ð42Þ
BSL ¼ γSL ASL
ð43Þ
BSV ¼ γSV ASL
ð44Þ
Figure 5. Contribution of each term in the free energy to the energy barrier for nucleation of a water droplet on fluid0 (dodecane) substrate and rigid 0 substrate with the same values of the interfacial tensions where T = 22 °C, PV = 2700 Pa, γLV = γw = 72.43 mN/m, γL V = γd = 25.3 mN/m, and γLL = wd γ = 53.7 mN/m: (a) BLV vs R, (b) BSL vs R, (c) BSV vs R, (d) BLaplace vs R. 10651
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For the fluid surface:
0
BLaplace ¼ ðP P ÞV V
L
ð45Þ
L
BLV ¼ γLV ALV 0
0
ðγL V γLL þ γLV Þ > 0
ð46Þ
0
BSL ¼ γLL ALL
ð47Þ
0
ð48Þ
BSV ¼ γL V πRU2 sin2 β
As can be seen in Figure 5, all contributions lead to easier nucleation on rigid surfaces except for the role of BLV. Actually, in nucleation on a fluid surface, due to the deformation of the surface beneath the drop, the area of the liquidvapor interface decreases to such a degree that it overcomes the increases in every other term and affects the overall result. Therefore, on the whole, nucleation of a drop on a fluid surface results in a lower energy barrier in comparison with nucleation on a rigid surface.
f
0
0
γLL < γL V þ γLV
ðA5Þ
The criteria A3 and A4 are the same as those given by Jarvis et al.,18 and the criterion A5 is the same as that given by Nepomnyashchy et al.16 In our paper, we chose dodecane as a fluid substrate for nucleation of water on it. In fact, we chose water and dodecane so that heterogeneous nucleation takes place providing a good model for comparing the barrier for nucleation on a “rigid” versus a “soft” substrate. 0 In our sample case: γLV = γw = 72.43 mN/m, γL V = γd = 25.3 0 mN/m, γLL = γwd = 53.7 mN/m: • Validation of criterion A3 0
γL V γLV ¼ γd γw ¼ 25:3 72:43 ¼ 47:13
mN mN 0 < γLL ¼ γwd ¼ 53:7 m m
• Validation of criterion A4 0
’ CONCLUSIONS We first explored the effect of contact angle on the barrier for nucleation on a rigid surface. It was found that the nucleation energy barrier increases with increasing contact angle. Next we considered two cases, one nucleation on a fluid surface and the other nucleation on a rigid surface with the same interfacial tensions as in the fluid case. It was shown that the energy barrier for heterogeneous nucleation on a fluid surface is lower than that for a rigid surface, so that heterogeneous nucleation will be easier on a fluid surface. In addition, by inspection of the contribution of different terms to the free energy, it was found that for nucleation on a fluid substrate the decrease in the liquid vapor area, ALV, decreases the free energy to the extent that it more than compensates for the energy increase due to the higher substrate-drop area and lower drop volume. It should also be mentioned that these are general conclusions for heterogeneous nucleation and that the values of interfacial properties were chosen for illustration only. ’ APPENDIX In order to find the criteria for surface and interfacial tensions so that the liquid lens can be formed, we use the contact angle and deformation angle relations which followed from solving eqs 38 and 39 for cos β and cos R or more directly from the cosine rule:16 0
L0 V 2
ðγ
Þ þ ðγ Þ ðγ Þ 2γL0 V γLV
0
ðγL V γLL γLV Þ < 0 0
0
• Validation of criterion A5 0
γLV þ γL V ¼ γw þ γd ¼ 72:43 þ 25:3 ¼ 97:73
ðγL V þ γLL γLV Þ > 0
f f
0
In order to find the criteria for nucleation of a sessile drop on a rigid substrate, we use the Young eq 18. Noting that 1 < cos θ < 1, the criteria would be γLV > γSV γSL
ðA6Þ
γLV > γSL γSV
ðA7Þ
Since in the rigid case we have S instead of L0 , by substituting γSV = 25.3 mN/m and γSL = 53.7 mN/m, we will have • Validation of criterion A6 mN m mN ¼ 72:43 m
γSV γSL ¼ 25:3 53:7 ¼ 28:4 < γLV
0
0
γLL > γLV γL V
ðA3Þ ðA4Þ
mN m mN ¼ 72:43 m
γSL γSV ¼ 53:7 25:3 ¼ 28:4
ðA2Þ
γLL > γL V γLV 0
mN LL0 mN > γ ¼ γwd ¼ 53:7 m m
• Validation of criterion A7
LL0 2
LV 2
mN LL0 mN < γ ¼ γwd ¼ 53:7 m m
ðA1Þ
We know that 1 < cos R < 1 and 1 < cos β < 1 should be met. By simplifying eqs A1 and A2 and by keeping in mind that surface and interfacial tensions are always positive, the following criteria are found: 0
¼ 47:13
0
ðγL V Þ2 þ ðγLL Þ2 ðγLV Þ2 cos R ¼ 2γLL0 γL0 V cos β ¼
γLV γL V ¼ γw γd ¼ 72:43 25:3
< γLV
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Phone: 1-780-492-7963. Fax: 1-780-492-2881. 10652
dx.doi.org/10.1021/jp202018e |J. Phys. Chem. B 2011, 115, 10646–10653
The Journal of Physical Chemistry B
’ ACKNOWLEDGMENT This research was funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the University of Alberta. J.A.W.E. held a Canada Research Chair in Interfacial Thermodynamics during this research. ’ NOMENCLATURE S Entropy U Internal energy T Temperature P Pressure V Volume Chemical potential of component i μi γ Surface tension Number of moles of component i Ni A Area R Radius j Dummy variable of integration θ Contact angle on a rigid surface β Contact angle on a fluid surface R Deformation angle R Universal gas constant G Gibbs free energy F Helmholtz free energy B Free energy given by eq 20 ’ SUBSCRIPTS U Upper L Lower c Critical ∞ Saturation 0 Reference state
ARTICLE
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