Langmuir 1998, 14, 4915-4928
4915
Equilibrium and Dynamics of Evaporating or Condensing Thin Fluid Domains: Thin Film Stability and Heterogeneous Nucleation Ashutosh Sharma Department of Chemical Engineering, Indian Institute of Technology at Kanpur, Kanpur 208016, India Received December 18, 1997. In Final Form: May 4, 1998 A continuum theory is presented for the equilibrium and dynamics of evaporating/condensing thin ( p0) with respect to a flat interface of the bulk liquid. Conversely, condensation is favored at concave surfaces/ menisci on completely wettable (Π > 0) surfaces, even for undersaturated vapors. Thus, the surface curvature and the wetting properties (nature of the disjoining pressure isotherm) both play important roles in determining the direction and the rate of phase change in thin liquid domains. For an arbitrarily curved surface of a nonuniform thin film, evaporation and condensation may both occur simultaneously at different portions of the film. As expected, eq 9 also gives the extended Kelvin equation, eq 7, at equilibrium (m ) 0). A simplified representation of eq 9 for nonisothermal evaporation was obtained by (44) Derjaguin, B. V.; Zorin, Z. M. J. Phys. Chem. (USSR) 1955, 29, 1010.
3µht + (3µKg/F)[p0 exp{-(VL/RT)(γfhxx + Π)} - pv] + [h3{γfhxxx + Πhhx}]x ) 0 (10) where subscripts x, t, and h are shorthand notations for differentiation, i.e., Πh ) ∂Π/∂h. Equation 10, which is a new result, shows that the chemical (second term) and mechanical (third term) effects are interrelated, and both together determine the spatiotemporal evolution of a thin film. A nondimensional version of eq 10 is given in the Appendix, which facilitates a compact representation of numerical results. The condition for the mechanical equilibrium is obtained by setting the total effective pressure gradient, (p - Π) ) [γfhxxx + Πhhx], to zero. This also makes the third term of eq 10 zero. The resulting equation can be integrated to give the following extended or augmented YoungLaplace equation of capillarity for axisymmetric or spatially periodic surfaces.
γfhxxC + Π ) K0
(11)
K0 is a constant of integration, which is readily seen to be equal to γf/R0 + Π0, where R0 and Π0 denote the radius of curvature and disjoining pressure at the surface point on the axis of symmetry where hx ) hxxx ) 0. The condition for mechanical equilibrium does not, however, guarantee the chemical (phase) equilibrium (m ) 0). The condition for the latter is the extended Kelvin equation, eq 7, which is recovered by setting the second term in eq 10 to zero. Equation 11 is identical to eq 8 for a particular choice of K0 ) -RT ln(pv/p0)/VL. Thus, establishment of a true chemical equilibrium automatically guarantees the mechanical equilibrium as well, viz. when m ) 0 from eq 9, the third term of eq 10 is identically zero. However, the reverse is not true, which points to the interesting possibility of attainment of quasi-mechanical equilibrium for a slowly evaporating/condensing liquid film/meniscus. This of course squares with a common observation that the equilibrium contact angles/shapes are obtained even when the liquid drop is not in true chemical equilibrium, and is evaporating slowly. The interplay between the flows resulting from the phase change and other mechanical factors depends crucially on the respective characteristic time scales inherent in eq 10. Finally, the behavior of a thin ( 0) with a shorter range attraction (SP < 0). The surface of a thin film is mechanically unstable when ∂Π/∂h > 0, e.g., for h > hC in system B. he defines the equilibrium thickness (Π ) 0) for the case of saturated vapors.
discussion to the prototypes of the disjoining pressure isotherms which display the characteristics shown in Figure 2. For weak electrical double layer effects, the total excess free energy per unit area (and the disjoining pressure) is the sum of the long-range apolar Lifshitz-van der Waals (LW) interactions, and the shorter range polar (P) interactions (e.g, hydrophobic attraction), which frequently show an exponential decay.1-4 The LW component of the excess free energy per unit area is given by ((-A/12)πh2), where A ) Aff - Asf (f ) film, s ) surface) is an effective Hamaker constant. Further, the effective Hamaker constant can be related to the LW component of the spreading coefficient, SLW, by A ) 12πd02 SLW, where d0 is an equilibrium cutoff distance (≈0.158 nm).2,45,46 The “soft” part of the total free energy per unit area can be represented as2,19-21
∆G ) ∆GLW + ∆GP ) SLW (d02/h2) + SP exp[(d0 - h)/lp] (12) where lp is a correlation length for the polar interactions which may of the order of 10 nm for the long-range hydrophobic attraction (SP < 0).1-3,45-48 In some illustrative calculations reported later, a lower bound for lp ) 0.6 nm is used.2 SLW and SP denote the apolar and the polar components of the total spreading coefficient, S ) SLW + SP, which can also be defined in terms of the apolar and polar components of surface and interfacial tensions.2,19-21 The principal advantage of representing the free energy in the form of eq 12 is that the macroscopic parameters of wetting, SLW and SP, can be determined from the simple macroscopic measurements of equilibrium contact angles.2,14,25 It is also important to note that the free energy and disjoining pressure of unstable fluid films (45) Van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Chem. Rev. 1988, 88, 927. (46) Van Oss, C. J. Colloids Surf. A 1993, 78, 1. (47) Ducker, W. A.; Xu, Z.; Israelachvili, J. N. Langmuir 1994, 10, 3279. (48) Basu, S.; Sharma, M. M. J. Colloid Interface Sci. 1996, 181, 443.
cannot be measured with any degree of certainty by direct methods, e.g., by atomic force microscopy. On the basis of an expression for ∆G, the disjoining pressure isotherm is easily obtained from Π ) -∂∆G/∂h. Figure 2B corresponds to the qualitative variation of the free energy and disjoining pressure in the case of hydrophobic attraction (SP < 0) combined with the van der Waals repulsion (SLW > 0) for systems displaying partial wetting. SLW > 0 (A < 0) and SLW < 0 (A > 0) in eq 12 correspond to the net van der Waals repulsion and attraction, respectively. Similarly, negative and positive values of SP correspond to the polar attraction (hydrophobic interaction)1-3,45-48 and repulsion (hydration pressure), respectively. The polar interactions vanish (SP ) 0) for films of apolar (hydrocarbon) liquids, for which the interactions are of purely van der Waals type. The LW repulsion results when the LW component of the solid surface tension is larger than the LW component of the surface tension of the film fluid.2 Thus, a long-range van der Waals repulsion, which promotes spreading and film stability,14,19,20 occurs for water on all moderate to high energy surfaces with surface tensions in excess of about 21.8 mN/m (≈ γwLW).2 Were it not for high negative values of SP for water on almost any surface,2,19-21,38 most surfaces would be completely water wet. As an example, SP for water equals -102 mN/m on any apolar (γsP ) 0, γswP ) γwP) surface. Increased polarity (hydrophilicity) of the surface promotes its polar interactions with water and makes SP less negative, thereby attenuating the hydrophobic attraction.21,38 Thus, the case of SLW > 0 and SP < 0, which corresponds to the disjoining pressure isotherm of Figure 2B, occurs for water on a majority of polymeric, inorganic (e.g., glass), and biological surfaces which display partial wetting.3,4,14,19-21 In contrast to the above case, the case of SLW < 0 and SP e 0, which corresponds to the disjoining pressure isotherm of Figure 2A, occurs for water on low energy surfaces (γsLW < 21.8 mN/m). In this case, the both the apolar LW force and the polar hydrophobic interaction engender partial wetting and promote dewetting of a thin film. Several examples of such systems are given elsewhere.19-21,38 Finally, SP vanishes for apolar liquids (e.g., octane, decane, diiodimethane). When the long range, “soft” part of the potential is purely attractive (SLW < 0, SP e 0) as in Figure 2A, the extremely short range Born repulsion (“hard” part) has to be explicitly retained to provide a cutoff for the LW forces which otherwise diverge as h f 0. The modified free energy in such cases may be written as23
∆G ) -A1[|SLW|(d02/h2) + |SP| exp[(d0 - h)/lp]] + B1|SLW|(d08/h8) (13) where the last term is derived from the pairwise summation of the r-12 repulsive term in the Lennard-Jones potential, and coefficients A1 ≈ 1 and B1 ≈ 27/256 satisfy the following requirements at the equilibrium intersurface distance, l0 ((31/2/4)d0 ≈ 0.137 nm): Π (l0) ) 0 and ∆G(l0) ) SLW + SP.2,23 The former requirement produces a minimum in the free energy at l0, and the latter ensures the proper correspondence between the microscopic and macroscopic definitions of the free energy, viz., the change in the free energy per unit area in reducing the film thickness from ∞ (bulk) to the cutoff distance must equal the spreading coefficient. Clearly, the “soft”, long-range part of the potential in eq 13 away from l0 is given by the form announced in eq 12. While formally the Born repulsion may also be included in the isotherm shown in
4920 Langmuir, Vol. 14, No. 17, 1998
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Figure 2B, its inclusion is not necessary to provide a shortrange cutoff for the LW forces (since SLW >0), unless the minimum in the free energy due to the long range forces occurs close to d0. The long-range potential, eq 12, of course remains largely unaffected. It may be noted that while the simple disjoining pressure prototypes, eqs 12 or 13, are the most convenient and physically meaningful models for simple fluids, the basic physics and qualitative conclusions obtained from their use remain unchanged as long as the shape of the isotherm resembles those shown in Figure 2, regardless of the physical origin of the interactions. Equilibrium Solutions Equation 7 or 8 together with an appropriate disjoining pressure isotherm provides the equilibrium shapes, which may be stable or unstable. Stability of an equilibrium solution may be judged either from eq 10 or from the free energy functional describing the total energy of the system. The former method provides additional details, e.g., dynamics and shape of the evolving structure, and will be followed here. Note that pve in eq 7 is now the actual vapor pressure pv with which the liquid is in chemical equilibrium. In what follows, we consider the stable equilibrium solution corresponding to the adsorption isotherm (pv < p0; flat surface, hxx ) 0), and unstable equilibrium solution in heterogeneous nucleation (pv > p0; convex meniscus, hxx < 0). Spatially periodic (unstable) equilibrium solutions for thin films also exist. These will be considered with the stability and dynamics of thin films, which, together with nucleation, form the two major themes of the present study. As will be shown, the physics of heterogeneous nucleation and thin film stability is intimately related to the understanding of adsorption isotherm and equilibrium wetting. Equilibrium adsorption and wetting are first discussed briefly with the help of potential, eq 12, which also facilitates a consistent interpretation of nucleation and thin film stability results in terms of the macroscopic parameters of wetting (SLW, SP, and contact angle). Adsorption Isotherm: Relationship with Macroscopic Parameters of Wetting. From eq 8, the adsorption isotherm is given by
RT ln(pv/p0) ) -VLΠ
(14)
For a completely apolar (SP ) 0) perfectly wetting (SLW > 0) liquid, one obtains the well-known3,5,44 relation for the thickness, he, of the adsorbed film, which in our notation is given by: RT ln(pv/p0)he3 ) 2VLSLWd02. The adsorption isotherm shows a monotonic increase to the point of condensation (he f ∞, pv f p0), and a greater spreading power (SLW) engenders thicker films. While eq 14 has also been used previously3,4,29 together with a somewhat poorly defined “structural” component of the disjoining pressure, we briefly revisit its results for the potential of the form given by eq 12. As is shown later, this will help us tie the results of nucleation and film stability with adsorption in a consistent way. A comprehensive review of the thermodynamic relation between the adsorption and disjoining pressure is given by Hirasaki,49 who also discusses the conditions for stability of adsorbed films. Although the adsorption isotherm and disjoining pressure (49) Hirasaki, G. J. In Contact Angle, Wettability and Adhesion; Mittal, K. L., Ed.; VSP Publishers: Amsterdam: 1993; p 183. Hirasaki, G. J. In Interfacial Phenomena in Oil Recovery; Morrow, N. R., Ed.; Dekker: New York, 1991; p 23.
Figure 3. Schematic representation of a typical adsorption isotherm for the system of Figure 2B. Branches AB and EF are stable, branch EC is unstable, and branch BC is locally stable but metastable to the process of nucleation.
are also related in the case of submonolayer adsorption,49 it should be noted that the results presented here are quantitatively meaningful only for the polymolecular films. However, all of the results presented here can be modified for submonolayers if the disjoining pressure isotherm is obtained by other models49 or by experiments. Our results for low vapor pressure adsorption on poorly wetted surfaces (-SLW/SP f 0) are included only for the qualitative considerations. For the case of partially wetting polar fluids, e.g. water (SLW > 0 and SP < 0; Figure 2B), a schematic representation of the adsorption isotherm is shown in Figure 3 based on eqs 12 and 14. The equilibrium solutions on branches AC and EF are at least locally (linearly) stable to infinitesimal perturbations, but the middle branch CE is unstable. This conclusion was also verified by direct numerical solutions of dynamic eq 10 starting with different initial conditions (results not shown).39,40 As is shown later, the segment BC is actually metastable, in that condensation can occur for supersaturated vapors by the mechanism of heterogeneous nucleation. However, the actual condensation may or may not be observed here in a finite time depending on the relative time scales for the nucleation and adsorption. Spontaneous condensation may be anticipated beyond point C, which is verified to be the case in the section on Heterogeneous Nucleation later in this paper. Points C and E are both nontrivial solutions of (dpv/dh) ) 0, and thus these points correspond to the two finite solutions of (dΠ/dh) ) 0, from eq 14. The reasons for the hysteresis of the adsorption are also clear from Figure 3, viz., the branch AC (with condensation beyond points B or C) is realized upon increasing of the vapor pressure, but decreasing of the vapor pressure traces the branch FE followed by a downward jump at point E to branch AB. Parts a and b of Figures 4 show the adsorption isotherms of water on a variety of surfaces based on the corresponding disjoining pressure isotherms derived from eq 12. Figure 4a shows the effect of increased wettability of the substrate as the LW component of the spreading coefficient becomes more positive. Increased wettability induces barrierless condensation at a lower degree of supersaturation; i.e., point C of Figure 3 moves to the left. At the same time, point E also moves to the left, meaning that much thicker stable adsorbed films on branch EF can exist at lower vapor pressures. For sufficiently high (positive) values of SLW in conjunction with low (negative) values of SP, points C and E merge to produce a monotonically increasing adsorption isotherm as in the case of perfectly wetting liquids. As an example, for lp ) 0.6 nm, (dΠ/dh) ) 0 has no finite solutions when (-SP) < 0.09SLW, and the adsorption shows a monotonic increase to ∞ as saturation
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Langmuir, Vol. 14, No. 17, 1998 4921
The first possibility is that a liquid drop is placed on an initially dry surface, the gas phase is devoid of the vapor, and the liquid is rather nonvolatile, e.g., polymer liquids and melts. In such cases, attainment of chemical equilibrium occurs over a very long time scale, and the condition for chemical equilibrium, eq 7, is not strictly satisfied for a relatively short period of observation of the contact angle. The drop can however attain mechanical equilibrium with a “hydrodynamic precursor”, viz., with a thin film which extends from the drop due to mechanical factors.4,7-14,50,51 For incomplete wetting, the (quasi) equilibrium thickness of such a film is given by Π(he) ) 0, and the resulting macroscopic contact angle can be obtained directly by integrating the mechanical equilibrium condition, eq 11.14 This procedure, as well as its other variants, give an augmented Young-Dupre equation for the contact angle, θ, as9-14, 49-51
cos θ ) 1 + ∆G(he)/γf
Figure 4. (a) Influence of SLW on the adsorption isotherm of water at 20° C. Curves 1-3 correspond to SLW ) 2, 10, and 30 mN/m, respectively. Other parameters are SP ) -10 mN/m and lp ) 0.6 nm. (b) Influence of SP on the adsorption isotherm. SLW ) 10 mN/m, and curves 1-3 correspond to SP ) -5, -10, and -25 mN/m, respectively.
is approached. Further, increasing the substrate wettability by decreasing the magnitude of the hydrophobic attraction (by making SP less negative) has the same effect on the adsorption isotherm as induced by an increase in SLW (results shown in Figure 4b). Since the adsorption isotherm depends on the individual values of both SLW and SP, adsorption isotherms of two different substrates with identical wettability (contact angle) may be very dissimilar (see the section on equilibrium contact angle). Finally, the use of eq 13 for completely apolar (SP ) 0), partially wetting (SLW < 0) hydrocarbon liquids shows submonolayer adsorption which may continue even for low to moderate degrees of supersaturation (results not shown). While this fact is qualitatively well accepted, the use of present theory for quantitative purposes in such cases is not justified. However, one important qualitative conclusion for the apolar fluids is that the branch BC of the isotherm continues for very high degrees of supersaturation before turning back, and thus the dominant mode of condensation for completely apolar liquids on partially wettable substrates is by nucleation, rather than by barrierless condensation. The latter is likely for water even at moderate supersaturations. Among other things, the above results help correlate the adsorption isotherm with the macroscopic equilibrium parameters of wetting. The case of complete wetting, e.g., water on extremely hydrophilic surfaces such as clean silica and glass, is not considered. Equilibrium Contact Angle of a Macroscopic Drop. From eq 7, a convex (hxx < 0), macroscopic (Π ) 0) liquid surface can never be in true chemical equilibrium with undersaturated vapors, since evaporation would always occur. In practice, a quasi-mechanical equilibrium may be attained when the drop recedes very slowly due to evaporation. There are two possibilities.
(15)
where ∆G(he) is the minimum in the free energy where Π ) 0. Equation 15 is also obtained for the special singular case of saturated vapors. ∆G(he) may also be referred to as the “equilibrium spreading coefficient” on the composite substrate made of the solid and its adsorbed equilibrium film.49 For monotonically increasing disjoining pressure isotherms (e.g., SLW and SP are both negative), the minimum in the free energy occurs at the cutoff separation, l0, and the classical Young-Dupre equation is recovered since ∆G(l0) ) S.14 A more detailed discussion may be found elsewhere.9-14,49-51 Also, the difference between the predictions of the augmented Young equation, eq 15, and the classical Young equation, cos θ ) 1+ S/γf, becomes small for moderate to large contact angles (>30°), where the “film pressure” defined as [∆G(he) - ∆G(l0)] is small. The second possibility is that either the surface is preequilibrated with the vapor phase or the liquid is moderately volatile and the time scale for adsorption is relatively short compared to the time scale for attainment of mechanical equilibrium. In this case the drop attains mechanical equilibrium with an adsorbed film of thickness, he, given by the adsorption isotherm, eq 23, viz., RT ln(pv/ p0) ) -VLΠ(he). The quasi-equilibrium contact angle (during the slow phase of drop-evaporation with mechanical quasi-equilibrium) is still given by eq 15 with a redefined he. Usually, adsorption increases the equilibrium contact angle compared to the case of a bare surface since usually for the systems of Figure 2b, ∆G(l0) ) S > ∆G(he); viz., ∆G(he) becomes more negative. For simple liquids, this is the only mechanism leading to the autophobic behavior. When two stable solutions exist for he (Figure 3), two distinct equilibrium contact angles from eq 15 also result. For such cases, the receding macroscale equilibrium contact angle (corresponding to higher value of he) should be close to zero since ∆G > 0 here (Figure 2b), but the advancing equilibrium contact angle (corresponding to the lower value of he) is finite. Finally, for extremely hydrophilic surfaces (e.g., fresh mica), ∆G may remain positive at all thickness; viz., the minimum in the free energy corresponds to ∆G > 0.57-59 In such cases, the (50) Brochard-Wyart, F.; di Meglio, J. M.; Quere, D.; de Gennes, P. G. Langmuir 1991, 7, 335. (51) Leger, L.; Joanny, J. F.; Rep. Prog. Phys. 1992, 55, 431. (52) Jameel, A. T.; Sharma, A. J. Colloid Interface Sci. 1994, 164, 416. (53) Van Der Hage, J. C. H. J. Colloid Interface Sci. 1983, 91, 385. (54) Scheludko, A.; Chakarov, V.; Toshev, B. J. Colloid Interface Sci. 1981, 82, 83.
4922 Langmuir, Vol. 14, No. 17, 1998
macroscale contact angle given by eq 15 is zero, but on microscales, two thin uniform equilibrium films can coexist.59 We consider here only the cases of partial wettability where eq 15 predicts finite macroscale contact angles. It may also be noted that the use of eq 15 for the disjoining pressure isotherm of the type in Figure 2B presupposes the existence of a contact line, e.g., where equilibrium is attained by spreading of a drop. If a continuous (without a contact line) thick film (satisfying Π > 0 and Πh < 0; Figure 2B) is initially deposited, it remains perfectly wetting and the apparent receding contact angle is zero. A finite angle can results only after the film becomes unstable due to evaporation (or nucleation) and breaks. This issue is also discussed later while considering the stability of thin water films on partially wettable surfaces. The above brief discussions of adsorption and wetting are intended primarily to interpret these phenomena in terms of the macroscopic parameters of wetting and to provide a background for the study of heterogeneous nucleation and the film stability, both of which have never been considered previously within the framework of eq 10. Heterogeneous Nucleation. Since condensation on perfectly wettable (Π > 0) surfaces occurs spontaneously, we consider the case of condensation by nucleation mechanism on partially wettable surfaces. The case of heterogeneous nucleation for supersaturated (pv > p0) vapors corresponds to the unstable equilibrium solutions of the basic eq 7 in the form of convex nanodrops (critical nuclei) coexisting with an adsorbed film of thickness he. It is also clear from the adsorption isotherm presented above that the process of nucleation is limited to supersaturations below a critical value (point C in Figure 3). The classical continuum theory of nucleation ignores the disjoining pressure effects as well as adsorption, and further assumes the nucleus to be of circular cross-section with a contact angle identical to that of a macroscopic drop.43 In effect, the classical theory is derived from the bulk thermodynamics, which neglects all finite size effects. This amounts to the neglect of Π in eq 7sa procedure which indeed would predict circular nuclei with contact angles nondifferent from their bulk values characteristic of macroscopic drops. By employing eq 7, we show that the approximations of the classical theory can seriously overestimate the energy barrier for nucleation and therefore greatly underestimate the rate of nucleation and overestimate the critical supersaturation. This is because the maximum height of critical nuclei is no more than a couple of nanometers at the most for significant (observable) rates of nucleation, which makes the finite size effects to be of paramount importance. For example, we have shown that the equilibrium contact angles of nanodrops can be significantly smaller than their bulk values for large drops due to the disjoining pressure and curvature effects.52 The shape of a nanodrop also becomes noncircular due to the presence of body forces (disjoining pressure) in the modified Young-Laplace equation.20,52 On the basis of a review of experiments, van der Hage53 concluded that condensation of water vapor on insoluble surfaces occurs at vapor pressures substantially smaller (55) Chakarov, V.; Scheludko, A.; Zembala, M. J. Colloid Interface Sci 1983, 92, 35. (56) Scheludko, A. Colloids Surf. 1983, 7, 81. (57) Elbaum, M.; Lipson, S. G. Phys. Rev. Lett. 1994, 72, 3562. (58) Lipson, S. G. Phys. Scr. 1996, T67, 63. (59) Samid-Merzel, N.; Lipson, S. G.; Tannhauser, D. S. Phys. Rev. E 1998, 57, 2906.
Sharma
Figure 5. Schematic profile of a critical nucleus on a substrate with adsorbed film of thickness he. xc denotes the position where h is within 0.1% of he.
than those predicted by the classical theory of nucleation. For water condensation on hexadecane, Scheludko and co-workers54 also found the value of critical supersaturation to be about 2.5 times smaller than that expected from the classical theory. A thin adsorption film of water on the substrate was thought to be responsible for the discrepancy,53-55 and an empirical “line tension” was invoked to reconcile the differences between experiments and the classical theory.55,56 In what follows, the extended Kelvin equation will be used to incorporate the finite-size effects and adsorption in a consistent way. The shape of a critical nucleus is obtained by integrating the extended Kelvin equation, eq 7, once to give
γf (1 + hx2)-1/2 + ∆G(h) - Ψh + C ) 0
(16)
where Ψ ) (RT/VL) ln(pv/p0), and C is a constant of integration. The actual profile, h ) h(x), can only be obtained numerically. This task is facilitated by noting that the far field (x f (∞) of the nucleus away from its center (x ) 0) is an adsorbed film of thickness, he, which is the lower stable root obtained on the branch BC of the adsorption isotherm, Ψ ) -Π, for pv > p0 (see Figure 3). The schematic representation of a nucleus merging with the adsorbed film is shown in Figure 5. Thus, eq 16 is to be considered together with the following conditions: (a) x f -∞, hx f 0, and h f he; (b) x ) 0, hx f 0, and h ) h0. Condition a with eq 16 gives
γf + ∆G(he) - Ψhe + C ) 0
(17)
Condition b above gives an identical relation as eq 17 with he replaced by h0. Combining these two relations to eliminate C gives an equation from which the unknown height, h0, of the nucleus may be obtained.
∆G(he) - ∆G(h0) + Ψ(h0 - he) ) 0
(18)
The constant, C, is then determined from eq 17. The left half of the symmetric equilibrium profile of a critical nucleus is now obtained from numerical integration of eq 16 starting at some (negative) xi, where h ) hi (he < hi < h0). Integrating in the forward direction for increasing x is continued until the height reaches h0 (where the slope also becomes zero). This point corresponds to x ) 0. Integrating in the reverse direction starting from xi traces the far field (foot) of the drop. Integration is stopped at a large negative x ()xc) where thickness becomes close (within 0.1%) to he. The contact angle, θmax, of the finite sized nucleus may be interpreted as the maximum slope of the profile. The other important parameters characterizing the shape are the maximum height (h0), radius of curvature (rc) at the apex, x ) 0, and the adsorbed film thickness (he). The radius of curvature at x ) 0 (h ) h0, hx ) 0) is obtained from eq 7 by noting that here, rc ) (hxx)-1, which gives rc ) [(Ψ + Π(h0)/γf]-1. As an example of the above methodology, we consider nucleation of water (T ) 20 °C, γf ) 72.8, γf LW ) 21.8
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Langmuir, Vol. 14, No. 17, 1998 4923
Table 1. Morphology and Nucleation Energy of Critical Nuclei of Water (T ) 20 °C, SLW ) 15, SP ) - 50; in mN/m) pv/p0
rC (nm)
he (nm)
h0 (nm)
θmax (deg)
∆F/∆Fc
1.03 1.10 1.15 1.20 1.25 1.30 1.34 1.35 1.38
18.25 5.66 3.86 2.96 2.42 2.06 1.84 1.80 1.68
0.22 0.23 0.24 0.25 0.27 0.29 0.31 0.32 0.36
9.58 3.10 2.11 1.56 1.19 0.90 0.70 0.65 0.47
52 39 32 25 19 13 8 7 2
0.94 0.77 0.61 0.45 0.30 0.16 0.07 0.052 0.005
mN/m, p0 ) 2.34 × 103 N/m2, and VL ) 0.018 m3/kmol) on a partially wettable surface characterized by SLW ) 15 mN/m and SP ) - 50 mN/m. These values correspond to a typical surface with γsLW ≈ 40 mN/m, and a macroscopic equilibrium contact angle of about 60°. Table 1 lists some of the important morphological parameters of critical nuclei for this system at different degrees of supersaturation. As expected, the size of critical nucleus becomes increasingly smaller at higher supersaturation as its radius (rc at x ) 0) and height (h0) decline. There is a concurrent increase in the thickness (he) of the adsorbed film which corresponds to branch BC of the adsorption isotherm in Figure 3. It is clear from Table 1 that the height of the critical nucleus approaches the adsorbed film thickness as point C in Figure 3 is reached, beyond which condensation occurs spontaneously, rather than by the mechanism of nucleation. Interestingly, the maximum slope (θmax) of the critical nucleus is substantially smaller than the corresponding macroscopic contact angle (≈60°) even at low degrees of supersaturation. As may be anticipated, the slope becomes zero as the point of barrierless condensation (point C in Figure 3) is reached. The size (radius, height, and volume) of the critical nucleus is also smaller than the prediction of the classical theory. Thus, assumptions of the classical theory regarding the shape and size of critical nucleus are in serious error unless the nucleus height is greater than about 10 nm, which is the case only at very low supersaturations on poorly wetted surfaces. These cases are, however, not of great interest, since the rate of nucleation is so low as to be unobservable on realistic time scales. Direct numerical solutions of the thin film eq 10 in the vicinity of the equilibrium solutions confirm that the critical nuclei discussed above are indeed unstable to the process of evaporation/condensation. For example, Figure 6 shows the growth of a slightly supercritical nucleus due to condensation for (pv/p0) ) 1.2. Figure 7 depicts the decay of a slightly subcritical nucleus due to evaporation for the same case. The growth/decay rate (but not the stability) depends on the mass transfer coefficient, the maximum theoretical value for which is assumed in all dynamic calculations here. The single most important parameter from any theory of nucleation is the barrier height for nucleation, ∆F, which determines the rate of nucleation which is proportional to exp(-∆F/kT). The energy required to create a critical nucleus is the difference between the energy of the critical nucleus (Figure 5) and the energy of the substrate with its adsorbed film. The free energy of a thin liquid domain in chemical equilibrium with its vapor is given by (A ) 2xc is the area covered on the substrate by the liquid)3,4,29
F/2 ) γf
∫0xcx1 + hx2 dx + Aγsf xc xc (RT/VL) ln(pv/p0)∫0 h dx + ∫0 ∆G(h) dx
(19)
Figure 6. Growth of a slightly supercritical nucleus of water by condensation. SLW ) 15 mN/m, SP ) -50 mN/m, pv/p0 ) 1.2, C1 ) 1.597, C2 ) -0.3794 × 10-2, and P ) 444. A thin condensate film emerges at t ≈ 8 due to the periodic boundary conditions used, which signify merging of the neighboring nuclei..
Equation 19 is the free energy functional associated with the extended Kelvin equation, eq 7. The first two terms are the interfacial energies associated with macroscopic liquid-vapor and solid-liquid interfaces of the nucleus, and the third term represents the free energy reduction in condensing the supersaturated vapor to form the nucleus. All of these terms are present in the classical theory,43 but the nucleus is assumed to have a circular profile. The last term, which provides finite-size corrections to the classical theory, gives the excess free energy due to intermolecular interactions. From the calculus of variations, it can be easily shown that the extrema of the free energy functional, eq 19, are indeed given by the extended Kelvin equation, eq 7, which governs the equilibrium profiles, including the shape of critical nuclei. The energy of substrate (with adsorbed film) is also obtained from eq 19 by noting that hx ) 0 and h ) he. The change in the total free energy upon formation of a cylindrical nucleus on an adsorbed film of thickness he is therefore given by
∫0xc{∆G(h) - ∆G(he)} dx] + xc [γf∫0 {x1 + hx2 - 1} dx] xc (RT/VL) ln(pv/p0)∫0 (h - he) dx
∆F/2 ) [
(20)
Profile of the critical nucleus, h ) h(x), is of course obtained from eq 16 as discussed previously. The upper limit of integration in the above should rigorously be replaced by ∞, but a finite cutoff, xc, is adequate for computations since h is very close (within 0.1%) to he beyond xc, and therefore ∆F f 0 for x > xc. The barrier height for nucleation according to the classical theory is a simplified case of eq 20 when the nucleus is assumed to be large enough for the excess interactions to be negligible (∆G ) 0). The nucleus profile, h(x), in this case is a circular arc with a constant radius of curvature, r, and contact angle, θ, which is given by the relation, ∆G(he) ) γf(cos θ - 1) from eq 15. The free energy change for the formation of a circular macroscopic drop of the classical theory is therefore obtained from eq 20 after some manipulations as
4924 Langmuir, Vol. 14, No. 17, 1998
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still remain, but other corrections delineated here are less significant than for water. As an example, we consider nucleation of octane vapors (T ) 20 °C, γf ) γfLW ) 21.8 mN/m, p0 ) 1.42 × 103 N/m2, and VL ) 0.1625 m3/kmol) on a low energy surface (γs ) 14.5 and SLW ) -8; both in mN/m). The barrier heights from classical theory at (pv/ p0) ) 1.2, 2, and 6 are higher only by factors of about 1.05, 1.25 and 2.1, respectively, compared to the results of the extended approach. This example further reinforces the idea that the classical approach is better suited to describe nucleation at low supersaturations and on poorly wetted surfaces with insignificant (submonolayer) adsorption. However, these cases are not of practical importance, since under these conditions, the critical nuclei are large and the rate of nucleation is insignificant. Stability and Dynamics of Evaporating Thin Films Figure 7. Decay of a slightly subcritical nucleus of water by evaporation. SLW ) 15 mN/m, SP ) -50 mN/m, pv/p0 ) 1.2, C1 )1.0701, C2 ) -0.5122 × 10-2, P ) 298. Eventually, an adsorbed equilibrium film is left on the substrate (curve 7).
∆Fc ) -(RT/2VL) ln(pv/p0)r2 (2θ - sin 2θ) + γf r(2θ - sin 2θ) (21) Thus, the radius, rc, of the critical (∂∆F/∂r ) 0) nucleus from the classical theory is nothing but the classical Kelvin relation43
rc ) γf[(RT/VL) ln(pv/p0)]-1
(22)
Substitution of eq 22 in eq 21 provides the barrier height from the classical theory for a cylindrical nucleus, which can be compared directly with the predictions of the full theory consisting of eqs 16-20, which are in effect the extended Kelvin equation together with its energy functional. The ratio of barrier heights as computed from the present theory and the classical theory is shown in the last column of Table 1 for different degrees of supersaturation. As expected, the ratio approaches unity as pv f p0, since the critical nucleus approaches macroscopic dimensions at low degrees of supersaturation, where the bulk thermodynamics of the classical theory is adequate. More importantly, the ratio rapidly declines to zero as point C in Figure 3 is reached, which signals the onset of barrierless condensation. Clearly, classical theory overestimates the barrier height for nucleation, and it can therefore seriously underestimate the rate of nucleation, especially at relatively higher supersaturations on more wettable substrates. While the results shown in Table 1 are for a substrate of moderate wettability (θ ≈ 60°), it can be verified based on eqs 16-20 that the classical theory becomes increasingly inadequate with increased wettability (decreased θ) and with increased hydrophilicity (less negative SP) of the substrate. As discussed previously in parts b and c of Figure 4, increased wettability induces barrierless condensation at lower supersaturation. The above discussion shows that the assumptions of the classical theory of nucleation are increasingly inadequate as point C of the adsorption isotherm is approached. As discussed previously in the section on Adsorption Isotherms, the point C is reached at extremely high values of supersaturation for completely apolar partially wetting liquids (eq 13; Figure 2A). This is the reason classical theory should work much better for apolar hydrocarbons at moderate supersaturations away from point C. Of course, limitations of a continuum approach
While the dynamics and stability of thin ( 0) combined with a hydrophobic attraction (SP < 0) as shown in Figure 2B. The longer range LW repulsion wins beyond a certain critical thickness, hC, and thus thicker films, once formed, remain mechanically stable and perfectly wetting since the condition,15-21 (∂Π/∂h) < 0, is satisfied beyond hC (Figure 2B). However, evaporation continues to thin the planar film until its thickness reduces below hC, at which time the free surface becomes unstable (∂Π/ ∂h > 0) due to dominance of the hydrophobic attraction. At this time the growth of surface instability together with the slow film thinning leads to dewetting by the formation of holes. For extremely hydrophilic surfaces (e.g., freshly cleaved clean mica without any ion adsorption), the minimum of the free energy (∆G) in Figure 2B may lie on the positive side, and thus eq 15 predicts complete wetting. The physics of instability (induced by nucleation) in these cases is presented recently elsewhere and is therefore not considered here.59 In what follows, we present a simplified linear stability analysis of the governing eq 10 to assess the influences of the rate of evaporation and surface properties on the film thinning and rupture. We seek the solution of the form, h ) h0(t) + h′(x,t), to the linearized version of eq 10. The base state (mean film thickness, h0) is time dependent due to evaporation, is a measure of the initial disturbance amplitude, and h′(x,t)
Thin Fluid Domains
Langmuir, Vol. 14, No. 17, 1998 4925
) h1(t) exp(ιkx), represents the (periodic) perturbation of wavenumber k from the mean film thickness. From eq 10, time evolution of the mean film thickness is given by
F(dh0/dt) ) -Kg[p0 exp{-(VL/RT)Π(h0)} - pv]
(23)
For undersaturated vapors (pv < p0), the rate of evaporation from a flat film decreases for positive disjoining pressures signifying a greater adhesion and spreading power of the liquid on the solid, as noted earlier. The growth of instability from the linearized version of eq 10 is governed by
F(dh1/dt) + (Kgp0VL/RT) exp{-(VL/RT)Π(h0)}[γk2 Πh(h0)]h1 ) 0 (24) The dominant length-scale of the instability, λ ) 2π/km, determines the number density of holes (NH ∝ λ-2) per unit area of the substrate.23,25 In the linear approximation, the dominant wavenumber, km can be found by solving eqs 23 and 24 for a large number of different values of k. The wavenumber which gives the minimum time for the film breakup is the dominant wavenumber. It is known that the problem of mode selection can be adequately addressed by the linear analysis for small initial amplitudes.18,20,21 However, even though analytical solutions to the coupled set of eqs 23 and 24 are not possible, the essential physics of mode selection can be examined by the use of quasi-steady approximation, which has been extensively used for thinning films starting from the seminal work of Vrij.60 When the rate of film thinning is slow compared with the growth rate of the instability, one can look at the solution of eq 24 for a fixed h0, and this solution changes quasi-steadily with slow changes in h0. This is a good approximation for low to moderately volatile liquids, e.g., water, especially when the vapor pressure is close to saturation. As is confirmed later by direct numerical solution of the full eq 10, the growth of instability (implied by eq 24) becomes explosive as the film thickness declines below hC, whereas the rate of evaporation is a relatively weak function of thickness near hC. This motivates a local analysis of the growth of instability at a nearly constant (slowly changing) mean thickness. The quasi-steady linear stability analysis gives the growth rate of the instability, ω, from h1 ) exp(ωt) and eq 24, which give a dispersion relation
ω ) [-γfk2 + Πh(h0)][h03k2(1 + V)/3µ + (Kgp0VL/FRT) exp{-(VL/RT)Π(h0)}] (25) where V ) 2m0(m0 + KgpV)VL/FVRT, in eq 25 is an additional contribution if vapor recoil (last term of eq 3) is also included in the analysis. m0 ) Kg[p0 exp{-(VL/ RT)Π(h0)} - pV], is the rate of evaporation of a planar film. Thus, vapor recoil has an additional destabilizing influence for unstable films satisfying, Πh(h0) > γfk2. However, it is easily verified that vapor recoil effect is entirely negligible (V , 1) except perhaps in the case of explosive evaporation corresponding to the film thinning rate ()m0/F) of about 10 m/s! We have also verified this conclusion based on direct numerical solutions of eq 3. The effect of vapor recoil will therefore be neglected in what follows. (60) Vrij, A. Discuss. Faraday Soc. 1966, 42, 23.
The fastest growing mode (∂ω/∂k ) 0) has a wavenumber km which determines the dominant length scale of the instability, λ ) 2π/km.
km2 ) {Πh(h0)/2γf} (3µKgp0VL/2FRTh03) exp{-(VL/RT)Π(h0)} (26) From eqs 25 and 26, the growth of instability (ω > 0) for an evaporating film can occur only when ∂Π/∂h > 0, as in the case of nonvolatile films.15-21 As expected, eqs 25 and 26 reduce to the corresponding equations for a nonevaporating film when Kg ) 0.15-21 The presence of an extra term (∝Kg) in eq 25 has a simple physical interpretation. The effect of surface curvature (and surface tension) is to increase the rate of evaporation from the convex (higher thickness) regions of the surface as compared to the concave (lower thickness) regions, where the rate of evaporation is decreased compared to a flat surface. This effect acts to decrease the growth rate, so that the stabilizing influence of the surface tension in eq 25 is further enhanced. The role of disjoining pressure can be similarly visualized. If disjoining pressure increases with the film thickness, the rate of evaporation becomes lower at the thicker regions of the film as compared to the thinner regions. Thus for Πh > 0, this effect leads to a greater destabilization (higher ω) of the free surface compared to the case of a nonvolatile film. The opposite is true for the case of Πh < 0, which enhances the stability (decay of perturbations). The stability of planar equilibrium solutions (eq 14) against mass loss/gain has already been discussed in the context of adsorption isotherms. Equation 25 predicts the morphological stability of the equilibrium solutions given by eq 14, as shown by the adsorption isotherm of Figure 3. From Figure 2B, it is readily verified that Πh < 0 for the smallest equilibrium solution (for both pv < or > p0), which falls on the rapidly ascending branch of Π as h f 0. Thus, all equilibrium solutions on the branch AC of the adsorption isotherm in Figure 3 are also stable against surface deformations. From Figure 2B, it can be similarly argued that Πh > 0 for the equilibrium solutions on the branch CE of the adsorption isotherm, which makes these equilibrium states unstable to shape changes, in addition to their instability by mass gain/loss. Finally, equilibrium solutions on the branch EF are stable due to Πh < 0. In what follows, we consider the cases where a single stable solution on branch AB exists, and therefore all thicker films are unstable (rather than metastable). Close to saturation where two stable equilibrium solutions exist (Figure 3), evaporation will merely thin the film to the higher equilibrium thickness (on branch EF of Figure 3) and the surface instability can only be induced by the process of “nucleation”, viz., by imposing a finite amplitude disturbance. On the basis of the quasi-steady approximation discussed above, the functional dependence of the length scale of the instability on the surface properties and the rate of evaporation may be determined to a first approximation by the following simple analysis, which has been used also for a similar problem of thinning of foam films.60,61 Close to hC, the rate of thinning due to evaporation is nearly constant since Π(hC) f 0 in eq 23 for hC > 10 nm and hence, the effect of disjoining pressure on the rate of evaporation remains negligible. Just below hC, the growth rate of instability is extremely small since Πh(hC) f 0 near hC. With a further decline in the film thickness due (61) Sharma, A.; Ruckenstein, E. Langmuir 1987, 2, 760.
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to evaporation, Πh becomes increasingly positive, and eventually the growth of instability becomes almost explosive in comparison to the evaporative thinning, since the growth rate is a stronger function of the film thickness. The overall process of film breakup may therefore be viewed as composed of two sequential steps: (a) evaporating thinning from thickness hC to some intermediate mean thickness hR without a significant concurrent growth of instability followed by (b) the growth of surface instability at hR leading to hole formation without a significant concurrent film thinning. The mean film thickness, hR, at the instant of film breakup, and the dominant length scale of the instability are obtained by minimizing the total breakup time starting from hC.60,61 The time taken to decrease the film thickness from hC to hR by evaporation is given, as a first approximation, by
tE ) (hC - hR)/E
(27)
where E ≈ Kg(p0 - pv)/F is the rate (volume flux, nm/s) of evaporation in the vicinity of hC. Further, at the mean thickness hR, the dominant growth coefficient is obtained from eq 25 with h0 ) hR, and the time of rupture is given by, tB ) ω-1 ln(hR/). Note that a true dry-spot does not form due to the film breakup, since an adsorbed equilibrium film of thickness he , hR is left on the substrate. The minimum in the total time, tE + tB, occurs at some intermediate thickness, hR, which can be found numerically from the above analysis. The dominant length-scale of the instability is then found from eq 26 with h0 ) hR. An entirely analytical estimate may also be derived for the case of slow evaporation where hR is only slightly smaller than hC.62 It is easily verified that for moderately volatile liquids (e.g., water) near room temperature, the effect of extra evaporation related terms on ω and km are entirely negligible for unstable thin films slightly away from where Πh ) 0. Thus, from eq 26, the length scale, λ, of the fastest growing mode of instability is approximately given by (subscripts R and C denote the function evaluated at hR and hC, respectively)
4π2λ-2 ) (∂Π/∂h)R/2γf ≈ -(∂2Π/∂h2)C(hC - hR)/2γf, for hR f hC
(28a) (28b)
The approximation eq 28b holds when hR is close to hC, which is the case when the growth rate of instability increases very fast with a decline in the film thickness, and/or the rate of evaporation is relatively slow near hC. Note also that (∂2Π/∂h2)C is negative near hC (Figure 2B). Further, the film breakup time by the growth of surface instability at hR (. he) is approximately given by15-24
tB ) 12µγf ln(hR/)hR-3[(∂Π/∂h)R]-2
(29a)
≈ 12µγf ln(hC/)hC-3[(∂2Π/∂h2)C]-2 (hC - hR)-2, for hR f hC (29b) where is the initial amplitude of disturbances, which is on the order of 0.1 nm for thermal perturbations. From the approximate analysis (eqs 27 and 29b), the minimum in the total time, (tE + tB), occurs at
(hC - hR)3 ) 24µγf ln(hC/)EhC-3[(∂2Π/∂h2)C]-2 (62) Sharma, A.; J Colloid Interface Sci. 1998, 199, 212.
(30)
and the dominant wavelength of the instability causing the hole formation is (from eq 28b)
4π2λ-2 ) [-3µγf-2 ln(hC/)EhC-3(∂2Π/∂h2)C]1/3; where (∂2Π/∂h2)C < 0 (31) Equation 31 is an approximate result concerning the initial number density (holes/area) of holes, NH ∝ λ-2, which open up in an evaporating film with a disjoining pressure isotherm of the type shown in Figure 2B. The hole density is independent of the initial thickness, but displays a weak dependence, NH ∝ E1/3, on the rate of evaporation. Evaporation may be enhanced either by lowering the vapor pressure (at fixed temperature) or by increasing the equilibrium vapor pressure (by increasing the temperature). The effect of surface interactions (surface properties and wettability) is contained in the critical thickness and in the curvature of the disjoining pressure isotherm at the critical thickness, viz., NH ∝ hC-1[-(∂2Π/∂h2)C]1/3. An increase in the range (lp) or the strength (-SP) of the hydrophobic interaction increases the critical thickness, hC, and simultaneously decreases the critical curvature, [-(∂2Π/∂h2)C]. Both of these factors decrease the initial number density of holes. This conclusion may appear to be paradoxical, since a stronger attraction causes stronger instability and greater hole density in nonthinning films at a fixed mean thickness.14-21,23-25 However, in the case of thinning films, the stronger hydrophobic attraction forces the onset (and conclusion) of instability at higher mean thickness, resulting in a longer characteristic length scale and lower hole density. Similarly, it can be verified that an increase in SLW increases the number density of holes in evaporating films, even though the substrate becomes more wettable. As discussed above, while the approximate analysis (eqs 28b, 29b, 30, and 31) is physically instructive, quantitatively it works better only for the case of strong instability in very slowly evaporating films where hR f hC. In other cases, the dominant wavelength should in general be found directly from eqs 23 and 24 by locating the minimum in the total time of rupture numerically. As discussed previously, another simplified but useful approach when hR is not close to hC would be to consider complete expressions 25 and 26, rather than eqs 28b and 29b, to arrive at the dominant wavelength. As an example of this more general approach, we consider the case of SLW ) 15 mN/m, SP ) -15.1 mN/m, and l ) 2.72 nm, which corresponds to water (γ ) 72.8 mN/m; µ ) 1 mPa‚s) on a partially wettable (θ ≈ 30°) typical (γLW ≈ 40 mN/m) surface with a moderate range of hydrophobic attraction. The critical thickness for the onset of instability is about 40 nm for this system. For E ) 0.1, 1, 10 and 100 (all in nm/s), the mean thickness at rupture (hR) equals about 35, 33, 30, and 26 (all in nm), respectively, and the dominant wavelength (λ) equals about 42, 24, 13, and 7 (all in µm), respectively. In contrast to the greatly simplified analysis, eq 31, the exponent q in the relation, λ-2 ) NH ∝ Eq, is no longer constant but varies between about 0.4 and 0.6 as E increases from 0.1 to 100 nm/s. Calculations were performed for a large number of combinations of SLW, SP, and lp in their feasible ranges. These revealed an interesting aspect, namely the length scale of the instability is rather more sensitive to E and hC but is almost independent of the individual values of parameters SLW, SP, θ, and lp, as long as their combination yields the same value of hC. For example, a nearly wettable (θ ≈ 1°) system, characterized by SLW ) 8 mN/m, SP ) -4.54 × 10-2 mN/m, and lp ) 5.576 nm (hC ) 50 nm),
Thin Fluid Domains
Figure 8. Evolution of surface instability in a nonevaporating water film. (a) The growth of initial perturbation (curve 1) leads to hole formation with a surrounding rim (curve 3). The merger of neighboring rims by hole growth leads to an equilibrium drop coexisting with an equilibrium thin film (curve 4). The periodicity interval, λ ≈ 0.2 µm, is renormalized to 2π. (b) Temporal evolution of the maximum and minimum film thicknesses. Parameters for a and b are as follows: h0 ) 4 nm, lp ) 0.6 nm, SLW ) 15.5 mN/m, SP ) -30.6 mN/m, C1 ) 0, and P ) 9391. Time is nondimensional (T; eq A4).
displays λ ) 27 and 54 µm, for E ) 0.1 and 1 nm/s, respectively. These values are not very different from those obtained for θ ≈ 30°, despite a greatly increased wettability. It is hoped that the proposed theory will provide a framework for the design and interpretation of future experiments on the vexing problem of the roles of surface properties and evaporation on the stability and pattern formation in thin volatile films. To assess the effect of evaporation on the surface instability and its evolution beyond the time of film breakup, we also numerically solved eq 10 together with the free energy formulation, eq 12. Simulations were performed for thin unstable water films on nonwettable substrates characterized by SLW > 0 and SP < 0. Equation 10 was solved in a domain of size, λ ) 2π/km, with periodic boundary conditions and a small amplitude ( ) 0.1) initial cosine perturbation. The successive central differencing in space with 60 grids was used together with the second order Everett method for the half-node interpolation. The resulting set of 60 coupled nonlinear ordinary differential equations were integrated in time with Gear’s method to account for the stiffness introduced by widely separated
Langmuir, Vol. 14, No. 17, 1998 4927
Figure 9. Evolution of surface instability in an evaporating water film. (a) The growth of surface instability with concurrent film thinning due to evaporation leads first to a quasiequilibrium drop structure (curve 3) as in Figure 8. However, evaporation continues to shrink the drop (curves 4 to 6) until an adsorbed film is left (curve 7). (b) Temporal evolution of the maximum and minimum film thicknesses. Parameters for a and b are as follows: h0 ) 4 nm, lp ) 0.6 nm, SLW ) 15.5 mN/m, SP ) -30.6 mN/m, pv/p0 ) 0.5, C1 ) 53.576, C2 ) -2.6807 × 10-4, P ) 9391, and λ ≈ 0.2 µm. Time is nondimensional (T).
length and time scales. Equation 10 was thus solved in its compact nondimensional form given in the Appendix. The typical behavior of nonevaporating and evaporating water films on a nonwettable substrate are summarized in Figures 8 and 9, respectively. The initial thickness ()4 nm) was chosen to be smaller than the critical thickness, hC for this case (≈6 nm). In the absence of evaporation (Kg )0; Figure 8a), the mean film thickness remains constant during the evolution of mechanical instability. The growth of instability leads to the formation of a hole surrounded by a rim of higher thickness (curve 3 of Figure 8a). The complete profiles of two neighboring holes are best visualized by looking at a periodic extension of the profile shown in Figure 8a. An ultrathin equilibrium flat film of thickness, close to that given by Π ≈ 0, is left behind on the substrate during the hole formation and its subsequent growth. The holes continue to grow until the merger of neighboring rims to produce an equilibrium drop (curve 4 of Figure 8a) which is also in equilibrium with the surrounding ultrathin film. The formation of a quasi-stable periodic structure consisting of two distinct morphological phases (drops and films) starting with a uniform film was earlier referred to as a “morphological phase separation”.19,20,52 Figure
4928 Langmuir, Vol. 14, No. 17, 1998
8b depicts the evolution of the maximum and minimum thicknesses for the nonevaporating film shown in Figure 8a. Parts a and b of Figure 9 show the salient features of the evolution when the effect of evaporation is also included. The maximum possible theoretical rate of evaporation with R ) 1 is chosen for illustration. Because of this reason, and also because a relatively low vapor pressure (pv ) 0.5p0) is chosen, the effect of evaporation is about the maximum possible. Comparison of Figures 8 and 9 reveals that the initial growth of instability until the formation of hole, as well as its dominant length scale, remain similar to the case of nonevaporating film. The thickness of the flat equilibrium film in this case is however smaller since it is close to the thickness of the stable adsorbed film given by the adsorption isotherm, eq 14. Also, unlike the case of nonevaporating film, the drop (curve 3 of Figure 9a) formed (largely) by the mechanical instability is no longer stable against the process of evaporation, since its profile cannot be a solution of the extended Kelvin equation. Evaporation continues to shrink the drop slowly (curves 3-7 of Figure 9a and Hmax in Figure 9b) until a flat adsorbed film is left (curve 7), which is the only stable solution of the extended Kelvin equation in this case. As discussed in the section on Wetting, the quasi-equilibrium contact angle of the evaporating drop is close to the prediction of eq 15 initially, but it decreases with the drop size due to finite size corrections52 at later times. An important observation (e.g., from Figure 9b) is that for unstable (h < hC, Πh > 0) thin water films (SLW > 0, SP < 0) on partially wettable substrates (θ > 0), the growth rate of hydrodynamic instability leading to the film breakup is very fast, whereas evaporation occurs on a longer time scale. Thus, the local analysis implied by eq 25 can be used as a good first approximation for the analysis pursued here. Complete numerical solutions of the nonlinear eq 10 starting with a thick (>hC) film are currently being pursued in our group to quantify the influence of various parameters on the instability. In contrast to the simple theory proposed here, however, it requires a large number of such (numerically) difficult calculations to assess the functional dependence of the instability on various parameters, which can vary over a large range. Summary A unified continuum theory for the equilibrium and dynamics of thin fluid domains on solid surfaces is presented and applied to the phenomena of heterogeneous nucleation and stability (rupture) of thin evaporating films. These phenomena are also correlated to equilibrium adsorption and wetting, and all of the results are thus consistently interpreted in terms of the equilibrium macroscopic parameters of wetting, viz., contact angle and the apolar and polar components of the spreading coefficient. In the case of heterogeneous nucleation, departures from the classical theory can be substantial for polar liquids such as water. For evaporating films, a general framework, as well as a simplified analytical treatment, is presented for the length and time scales of the surface instability.
Sharma
Acknowledgment. Help received from Raman, S. Salaniwal, R. Khanna, and S. Padmakar in numerical computations is gratefully acknowledged. Correspondence with S. G. Lipson and D. S. Tannhauser kept alive the author’s interest in the problem of evaporating films. This work was partially supported by the Indo-French Centre for the Promotion of Advanced Research/Centre FrancoIndien Pour la Promotion de la Recherche, for which interactions with G. Reiter are gratefully acknowledged. Appendix The evolution eq 10 together with the free energy expression 12 can be transformed to a convenient nondimensional form by introducing the following scalings.
H ) h/h0, l ) lp/h0, d ) d0/h0, ν ) µ/F (A1) C ) 3Fν2/γfh0, B ) -2SLWd2h0/Fν2, P ) -SP/6d2l2SLW (A2) C1 ) Kgp0h0/FνB2C, C2 ) 3BVL Fν2/RTh02, R1 ) pv/p0 (A3) X ) (|B|C)1/2 x/h0, T ) B2 Cνt/h02
(A4)
The evolution equation in its nondimensional form is
HT + C1 exp[C2 (sgn(SLW) HXX + Φ] - C1R1 + [H3(HXXX + sgn(SLW)ΦH HX)]X ) 0 (A5) where
ΦH ) -H-4 + P exp{(d - H)/l}, Φ ) H-3/3 Plexp{(d - H)/l} (A6) Subscripts are used as a shorthand notation for differentiation, viz., ΦH ) ∂Φ/∂H, HX ) ∂H/∂X, etc., and the function sgn(SLW) denotes the sign of SLW, viz., sgn(SLW) ) +1 for SLW > 0 (the case considered here for water films) and sgn(SLW) ) -1 for SLW < 0. Linear stability analysis of eq A5 with H ) 1 + exp(ιKX + ΩT) gives the nondimensional dominant wavenumber (nondimensional counterpart of eq 26) as
2 sgn(SLW)Km2 ) C1C2 exp(C2Φ) + ΦH
(A7)
Φ and ΦH in eq A7 are evaluated from eq A6 at H ) 1. Finally, the dominant wavelength in the (nondimensional) X coordinate is Λ ) 2π/Km. In nondimensional simulations reported in this paper, eq A5 was solved. For simulations related to the nonlinear evolution of thin films, the periodicity interval was chosen as Λ. LA971389F
