Langmuir 1993,9, 861-869
861
Relationship of Thin Film Stability and Morphology to Macroscopic Parameters of Wetting in the Apolar and Polar Systems Ashutosh Sharma Department of Chemical Engineering, Indian Institute of Technology at Kanpur, Kanpur 208016, India Received October 8, 1992
The total free energy of a thin fluid film, sandwiched between a substrate and a fluid phase, is derived from the apolar (Lifshitz-van der Waals) and the polar (acid-base) interactions. While the instability and rupture of the thin f i i engenderedby the apolar forces have been extensively studied,the polar interactions cannot be neglected unless the film fluid and at least one of the bounding media are completely apolar. For polar systems (e.g., water fiis), the total free energy of the film and its second derivative (forceper unit volume) are shown to be related to the macroscopic parameters of wetting (i.e., the apolar and polar spreadingcoefficientsand contact angle of macrodrops). Threshold of the film instability is derived from the Young-Laplace equation modified by intermolecular pressure, and the growth rate of instability is obtained from Navier-Stokes equations. Regions of the film stabilitylinstability are determined from characterizationof all possible variationsof intermolecular forces with the film thickness,with components of spreading coefficients and macroscopiccontactangle. For systemsthat are completely apolar,completely polar, or when components of spreading Coefficients are of the same sign, the film rupture is guaranteed whenever the macroscopic drop displays a finite contact angle. Interesting possibilities occur when the apolar and polar spreadingcoefficientsare of opposite signs: (a)the thin film may be unstable even though the corresponding macroscopic drop shows complete wetting; (b) the film may be stable despite a finite contact angle. Conditions under which growth of instability leads to the f i i breakup, or to spatially nonhomogeneousfiis,are discussed. Implications of result in rupture of macroscopicfilms, heterogeneous nucleation, and the equilibrium film pressure are indicated. Introduction Fundamental investigationsl-ll of stability of thin fluid films on surfaces have been motivated by such diverse applications as flotation of particles, surface-boiling/ condensation,12J3adhesion of deformable particles (e.g., bubbles, ceUs,7J4drops) to surfaces,multilayer adsorption/ film pressure, engulfmentlrejection of particles at interfaces, kinetics of precursor films in wetting,l5!16 heterogeneous nucleation, and many more. Instability of thin biological films6JJ4(e.g., aqueous and mucous filmsg on the cornea) have also been thought to derive from excess intermolecular interactions that are characteristic of ultrathin films with mean thicknesses less than about 100 nm. In some applications, the initial thickness of the film may be macroscopic, but instabilities derived from macroscopic phenomena (e.g., Rayleigh-Taylor instability, Helmholtz instability, thermocapillarity, etc.) lead to growth of perturbations until the film thickness attains microscopic dimensions at some localizedspots. For films bounded by rigid substrates, fluid mechanical considerations showthat the viscous resistance to the flow becomes arbitrarily large, and the rate of decrease in the film (1) Sheludko, A. Adv. Colloid Interface Sci. 1967, 1, 391. (2) Vrij, A.; Overbeek, J. Th. G. J. Am. Chem. SOC.1968,90, 3074. (3) Felderhof, B. U. J. Chem. Phys. 1968,49,44. (4) Ruckenstein, E.;Jain, R. K. J.Chem. SOC.,Faraday Trans.2 1974, 70, 132. (5) Jain, R. K.; Ruckenstein, E. J. Colloid Interface Sci. 1976,54,108. (6) Maldarelli, C.; Jain, R. K.; Ivanov, I.; Ruckenstein, E.J. Colloid Interface Sci. 1980, 78, 118. (7) Dimitrov, D. S. Prog. Surf. Sci. 1983, 14, 295. (8) Williams, M. B.;Davis, S. H. J. Colloid Interface Sci. 1982,90,220. (9) Sharma, A.; Ruckenstein, E.J. Colloid Interface Sci. 1985,106,12. (10) Sharma, A.; Ruckenstein, E. Langmuir 1986,2,480. (11) Gupta, A.; Sharma, M. M. J. Colloid Interface Sci. 1992,149,407. (12) Bankoff, S. G. J. Heat TramferlTrans. ASME 1990, 112, 538. (13) Neogi, P.; Berryman, J. B. J. Colloid Interface Sci. 1982,88,100. (14) Gallez, D.; Coakley, W. T. Prog. Biophys. Mol. Biol. 1986,48,155. (15) de Gennes, P. G. Rev. Mod. Phys. 1985,57, 827. (16) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971,35, 85.
thickness becomes arbitrarily slow, as the film attains microscopic dimensions and continues to thin.lJJG18 A true rupture of the film under these circumstances occurs only when the excess intermolecular interactions can destabilizelocallythin domains of the film. This is possible because excess intermolecular forces (integrated over the film) show a rapid buildup with decline in the film thickness. An important example of such systems is the flow of liquid films on inclined surfaces, where the primary instabilities of the (macroscopic) film are caused by a combination of factors: flow, curvature of support, thermocapillarity, shear stresses in the adjoining phase, and viscous stratificati~n.l~-~l An understanding of conditions under which long range (integrated)intermolecular forces can engender formation of ((dry” spots is important in trickle bed reactors, thin film heat and mass transfer devices, and supported area equipment for distillation.22 AU of the previousstudies of f i i stability have,however, considered only the apolar films bounded by the apolar media, where instabilityis engendered only by the Lifshitzvan der Waals (LW) forces. This is a serious limitation in that f i i s of polar liquids (e.g., water) on polar substrates are more frequently encountered in practice. The effect of polar interaction^,^^-^^ variously disguised as the hydration pressure, structural forces, hydrogen bonding, (17) Joo, S. W.; Davis, S. H.; Bankoff, S. G. J. Fluid Mech. 1991,230, 117. (18) Pumir, A.; Manneville, P.; Pomeau, Y. J . Fluid Mech. 1983,135, 27. (19) Shalang, T.; Sivashinsky, G. I. J. Phys. (Paris) 1982, 43, 459. (20) Benney, D. J. J. Math. Phys. 1966,45,150. (21) Chang, H.-C. Chem. Eng. Sci. 1986,41, 2463. (22) Berg, J. C. In Surfactants in ChemicallProcess Engineering;
Wasan, D. T., Ginn, M. E., Shah, D. O., Eds.;Surfactant Science Series; Marcel Dekker, Inc.: New York, 1988; Vol. 28, p 29. (23) Israelachvili, J. H. Intermolecular and Surface Forces; Academic Press: New York, 1985. (24) Van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Adv. Colloid Interface Sci. 1987, 28, 35.
Q743-7463/93/24Q9-0S6~~Q4.o0/0 0 1993 American Chemical Society
Sharma
862 Langmuir, Vol. 9, No. 3, 1993
hydrophobic interactions, etc., are clearly important for the polar systems. As an example,roughly 70 7% of energy of cohesion (surface tension) of water is derived from the polar "acid-base" interacti0ns,2~and the rest is attributable Fluid tluid ( 3 1 to the apolar LW interactions. Thus, macroscopic pheI nomena involving polar interfaces (e.g., surface and interfacial tensions, contact-angles, etc.), as well as the excess energy of interaction for thin films,depend crucially on the strength of polar interaction^.^^^^ Examples of Figure 1. Schematic presentation of the film (3) bounded by a the polar liquids and substrates abound; much attention substrate (1) and a semi-infinitefluid (2). The instability of the free interface leads to the film deformation. has been devoted to measurements of polar interactions by direct force m e a s ~ r e m e n t sand ~ ~ also * ~ ~by? less ~ direct, but more convenient, measurements of macroscopic paof interfacial energies between semi-infinite phases. rameters (e.g., phase interface tensions, contact angles) In what follows, the sum of the apolar and polar forces that result from (integrated) polar interactions.2k34Since per unit volume of the liquid film is introduced in the the nature (attractive or repulsive), strength, and funcNavier-Stokes (N-S) equations as "excess" body force^.^^^ tional dependence of the polar interactions bear no The critical conditions for the growth of infinitesimal relationship to the corresponding quantities for the LW perturbations are identified from the resulting Younginteractions, the stability of a thin polar film may be Laplace equation of capillarity modified by body forces. profoundly altered, both qualitatively and quantitatively. The growth rate of the fastest growing perturbations and An objective here is to determine conditions for the film an estimate for the time of rupture (timescaleof instability) instability and its growth rate under combined action of are obtained from a linear stability analysis of N-S the apolar and polar interactions. Whether the growth of All of equations in the "thin-film" approximati~n.~Jj?~-~O intial instability leads to the film rupture, or merely to a the results for stability of thin f i are interpreted in steady corrugated film, is a delicate issue, which is also terms of wetting behavior of corresponding thick drops, addressed. It is known from the nonlinear a n a l y s e ~ ~ J ~ 7 ~ ~ which is characterizedby values of the spreadingcoefficient that the secondpossibilityof a steady,spatiallynonuniform and contact angle. Conditions are identified for true film is ruled out for the apolar systems,and a true rupture rupture of the f i ,as well as for the formation of structures of the film, leadingto microdrops,is guaranteed whenever consisting of microdrops in equilibrium with intervening the effective Hamaker constant is positive. A complete thin films. Finally, an extensivenumerical searchis carried classificationof qualitativelydistinct types of film stability/ out to divide the parameter space of the problem in instability characteristics is pursued for understanding of different regions; where each region corresponds to a film stability in different system. For example,theoriesl-ll qualitatively distinct stability behavior of the film. In incorporating only the LW forces predict the film to be this way, the instability of the free interface of the film unconditionally stable whenever the effective Hamaker is correlated with such parameters as the mean film constant is negative, but the polar interactions may lead thickness, the apolar and polar spreading coefficients,and to its destabilization,and even rupture, when the substrate the contact angle. In particular, conditions are established is rendered nonwettable due to the polar interactions (e.g., under which a thin film is stable (wettingbehavior for the the polar cohesive forces in the thin film). This brings us film) but its drop shows a finite contact angle, and also to the question of possible relationships between the conditions which imply instability of the film even when stability of microscopic films and wetting behavior of the the contact angle of macroscopic drop is zero (wetting correspondingmacroscopic drops. While previous studies drop). Implications for some phenomena involving thin have not addressed this question, some relation between films (e.g., heterogeneousnucleation,wettingby thin f h , the stability of thin films and the degree of wetting by rupture of macroscopic films, and equilibrium film presmacroscopic drops may be anticipated. This is because sure) are noted. the macroscopic parameters (surface and interfacial tensions)governingthe wettability (contact angle) also derive from the sum of intermolecular interaction^.^^^^^,^^ The Theory energies of interactions in the thin film are but remnants Consider an unbounded ultrathin fluid film (3)of mean thickness, ho, less than about 100 nm, which is sandwiched (25) Van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Chem. Reu. 1988, between a solid (1) and a bulk fluid (2) phase (Figure 1). 88, 927. (26) Van Oss, C. J.; Good, R. J.; Chaudhury, M. K. Langmuir 1988,4, For pure films on uncharged substrates, only the apolar 884. (Lifshitz-van der Waals) and the polar (acid-base) in(27) Van Oss, C. J. J. Dispersion Sci. Technol. 1991,12, 201. termolecular interactions need to be considered. The (28) Van Oss, C. J.; Giese, R. F.; Costanzo, P. M. Clays Clay Miner. 1990, 38,151. stability and dynamics of the film are then described by (29) VanOss,C.J. InBiophysicsofthe CellSurface; Glaser,R.,Gingell, Navier-Stokes equations with the inclusion of body forces D., Eds.; Springer-Verlag: Berlin, 1990; p 131. clue to excess intermolecular interactions, that are not (30) Sharma, A. J. Dispersion Sci. Techno!. 1992, 13, 459. (31) Janczuk, B.; Kerkeb, M. L.; Bialopiotrowicz, T.;Caballero, F. G. accounted for in the macroscopic definition of interfacial J . Colloid Interface Sci. 1992, 151, 333. tensions. The Navier-Stokes equations for the film are (32) Van Oss, C. J.; Chaudhury, M. K.; Good, R. J. Sep. Sci. Techno!. (for 0 Iz Ih(x,t)) 1989,24, 15. (33) Costanzo,P. M.; Giese, R. F.; van Oss, C. J. J. Adhes. Sci. Technol.
1990, 4, 267. (34) van Oss, C. J.; Good, R. J.
J. Protein Chem. 1988, 7, 179. (35) Israelachvili, J. H.; Adams. G. E. J. Chem. SOC.,Fardav Trans. 1978, 74,975. (36) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991,353,
--". 9.19
(37) Sharma, A,; Ruckenstein, E. J. Colloid Interface Sci. 1986,113, 456. (38) Fowkes, F. M. J. Phys. Chem. 1962,66, 382.
(av/at)
+ (V.V)V = -(l/p)
vp
+ Y v2v- (l/p)Vc$
(1)
where V is the velocity vector and 4, p , Y, p, and t are the intermolecular potential per unit volume, pressure, kinematic viscosity,film fluid density,and time, respectively. The term V 4 represents the excess intermolecular forces per unit volume of the film.'-10
Stability of Thin Fluid Films
Langmuir, Vol. 9, No.3, 1993 863
The total free energy of interaction (per unit area) of two bulk phases (1 and 2) separated by a film of fluid 3 is the sum of the apolar (LW) and polar (P) energies of interactions.23-29-38
= Spexp[(do- h)/ll (9) where Spis the polar component of spreading coefficient defined as
(2) AGY$(h) = AGkg(h) + AGY3,(h) For nonretarded Lifshitz-van der Waals interactions, the excess free energy per unit area is represented as
yiF are the polar components of interfacial tensions and
GLw 132 = -A/12rh2 (3) where h is the film thickness and the effective Hamaker constant, A, is defined in terms of the Hamaker constants for various binary interactions. A =A33 + A,,-A13-A23 (4) The proper cut off distance for the film thickness in eq 3 is the equilibrium distance, do, which arises due to the extremely short range Born repulsion+-12 term in the Lennard-Jonespotential). The best estimate for do is 1.58 A,25and for h < do, the Born repulsion may be replaced by a vertical rise in the potential to infinity (hard sphere approximation). Thus, in bringing two bulk materials 1 and 2 from infinity to the equilibrium separation distance of do, the LW component of free energy (eq 3) changes as follows The same change in the free energy may also be represented in terms of interfacial tensions as
where y$w is the LW component of interfacial tension between phases i and j. Clearly, the right hand side of eq 6 is the LW component of spreading coefficient, SLwof the film liquid 3 on support 1 in presence of bounding and 7 ) : fluid 2. If the bounding fluid (2) is a gas, become the LW components of surface tensions of the solid and the film liquid, respectively. In view of the two equivalent definitions of the free energy of adhesion (eqs 5 and 6) the excess free energy due to LW forces (eq 3) may be rewritten in a form that clearly brings out its dependence on a macroscopic parameter-the LW component of spreading coefficient.
ykr
(7)
When SLw is positive, adhesion is impeded by LW forces (repulsion), otherwise it is encouraged (attraction). The apolar, Lifshitz-van der Waals component of spreading coefficient is determined from eq 6 in conjunction with the Good-Girifalco-Fowkes combining rule39 which gives
AGf32
1 is a correlation length for the polar fluid. For water, 1 appears to be in the range of 0.2 and 1nm, and an estimate of the best value is about 0.6 nmSmThe reason AG& must equal Spat contact (h = do) is clear from the logic employed in derivation of eqs 5 and 6 for LW energies. The conclusion is, in fact, independent of the functional form of the free energy, AG132(h). Spmay be evaluated either from direct force measurements or, more conveniently, by a model of polar interactions used in conjunction with contact angle measurements. One useful model,2&Bfrom which consistent values of polar interfacial tensions (and therefore, Spfrom eq 10)can be determined, defines two independent parameters of surface polarity: an electron donor (proton acceptor) parameter Ti-, and an electron acceptor (proton donor) parameter Ti+. The polar component of surface tension due to "acid-base" (AB) interactions is given by where yi+ and yi- for a substance may be obtained from contact angle measurements.2-1 For water, y+ = y- = 25.5 mN/m, which gives the expected value of the polar component of water surface tension to be 51 mN/m. The polar component of the spreadingcoefficientis then obtained in terms of these acid and base parameters of the three materials involved (1, 2, and 3) as2&S sp
+
= S A B = 2[(y3/2((y131/2 ( y p - (yJ1/2) tyJ1~2((yl+)1/2 + (y2+)1/2- (YJ1/2)
+
- (Yl+YJ1/2
-
(r&+)1/21 (12) If either the bounding medium (2), or the substrate (l), is apolar (y1+= yi- = 0; i = 1 or 2), the expression simplifies considerably. SABcan be positive or negative, corresponding respectively to "hydration pressure" and "hydrophobic interaction". It is always negative when the film is apolar, but bounding media have conjugate types of polarity, and also when the film fluid (3) is polar, but bounding media are apolar. Positive SAB is found for water on intensely hydrophilic polymers and b i o s u r f a c e ~ . ~ ~ 3 ~ Acid-base parameters (y+ and y-) for many liquids and solids are known or may be determined by contact-angle goni0metry.~'3~ A summation of expressions 7 and 9 gives the total free energy of interaction per unit area, and a differentiation yields the energy (4) per unit volume of the film.
4 = -2SLW(d;/h3) - (Sp/l)exp[(do- h)/ll (13) It is now well-knownl-10that the film interfaces become unstable when interactions are solely of the LW type and Clearly, SLwmay be positive or negative, depending on the effective Hamaker constant is positive. In such an relative magnitudes of LW componentsof surface tensions, event, growth of interfacial deformations leads to the film ylLw,yzLW,and ~ 3 The ~ effective ~ . Hamaker constant breakup. However, polar interactions can be neglected than has a sign opposite to that of SLw. only when the film fluid, as well as at least one of the The polar component of the excess free energy is bounding media, are completely apolar. In this case, the obtained from an integration of the polar component of polar component of the spreading coefficient (eqs 10 and the pressure (e.g., hydration pressure), which shows an vanishes. exponential decay with film t h i ~ k n e s s . ~ ~ ~ 2 3 ~ ~ 5 ~12) ~~ ~ ~ ~ ~ ~ Since 5 , 3 6the system of the thin film with ita bounding phases is an isolated (closed)system, indefinitely sustained temporal oscillations of the film surfaces are (39)Good, R.J.; Girifalco, L.A. J . Phys. Chem. 1960,64, 561.
Sharma
864 Langmuir, Vol. 9,No. 3, 1993 ruled out. Such oscillations may be possible only for an ideal case of frictionless invicid fluid, or in the event of exchange of energy with external sources. Thus, the condition for the threshold of the film instability is directly obtained from N-Sequations (eq 1)by setting velocities to zero. This threshold (critical) condition corresponds to the neutrally stable waveforms that separate regions of stability from instability in the parameter space. From eq 1,the film interface becomes neutrally stable when the pressure forces exactly counter the intermolecular forces (per unit volume), viz. P* + 4, = 0
(14)
and (15) Pz + 42 = 0 where subscripts denote differentiation. The above two conditions for equilibrium are to be considered in conjunction with the equilibrium pressure jump boundary condition at the film interface (theYoungLaplace equation of capillarity)
+
p, - p = 723hxx(1 hx2)-3/2at z = h
(16) where POis pressure in the bounding phase 2 (assumed constant) and 723 is the film interfacial tension with the bounding fluid. Integration of eq 15 shows that the sum of pressure and intermolecular potential is constant across the film thickness P(Z) + 4(z) = P(h) + 4(h) (17) where p(h) is given by eq 16 and +(h)is obtained earlier (eq 13). Substitution of these expressions in eq 14 gives a modified Young-Laplace equation for the thin film that incorporates intermolecular interactions
4%- (d+/dh)h, = 7 2 3 [h,,(l + h, I, (18) where (ddldh) is evaluated from expression 13 as
at the mean film thickness is negative, viz. (d4/*)b+o < 0
(21)
when written out in full, this condition is (from eq 19)
+ SP(h;/l2)exp[(do- h,)/lI < 0 (22) 6SLW(d,2/h,2) whenever the above condition is satisfied, the film is unstable to an infinite set of wavenumbers, k < k,. The initial rate of growth of unstable infinitesimaldisturbances, however, depends on the details of the dynamic model. The simplest model that illustrates all of the relevant physics is based on the approximation that the wavelengths of disturbances are large compared to the film thickness ( k b 0 y3LWor rlLW and yZLw < y3LW (28)
Any of the above two conditions also guarantees a positive effectiveHamaker constant for the system (eq 5). The Young equation for Sp = 0 simplifies to
sLW = yZ3(cose - 1)
for sLW 0). This corresponds to a type 4 curve of Figure 2. The domain of film instability is then confined to film thicknesses falling in the range of h1 to hz. As shown in Figure 5, the window of instability of film enlarges as the ratio JSP/SLwI increases. This is in accord with the
Langmuir, Vol. 9, No. 3, 1993 867
Stability of Thin Fluid Films
10 P -A SLW 1
047
-
1
I
Figure 5. Regions of film stability and instability when the apolar and polar spreading coefficientshave differentsigns. The curve represents the boundary where #h changes sign (see is less than about diagrams 2-4 of Figure 2). If the ratio ISp/SLwI 0.1, r#Ih maintainsthe same sign for all film thicknesses(diagrams 1 and 3 of Figure 2).
macroscopic behavior, since for Sp < 0, the polar interactions resist the drop spreading. In contrast to the above, f$h is positive in the range of hl to h2 when Sp> 0 (or SLw < 0). This corresponds to type 2 curve of Figure 2. In such an event, the initial instability of the film occurs either when it is thicker than h2 or when it is thinner than hl. However, instability for h > h2 is usually weaker, since magnitude of & becomes small at large thicknesses. Films with thicknesses in the range of hl to h2 are stable against infinitesimal disturbances due to positivity of $Jh. Since spreading of macroscopic drops is assisted by polar interactions when SP > 0, an increase in the ratio ISp/SLw( enlarges the window of film stability in this case. An important aspect of film stability for the polar systems is that, unlike the apolar systems,the sign of total spreading coefficientdoes not determine the stability characteristics for all film thicknesses. As an example consider the case where
+
S = Sp SLw> 0; Sp< 0 andSLW> 0 The above conditions also imply that the ratio Sp/SLw is less than 1. The Young equation for S > 0 predicts a complete spreading of the macroscopic drop (0 = 01, resulting presumably in a flat film of molecular thickness. However, from Figure 5 and curve 4 of Figure 2, it is clear that the film becomes unstable once ita thickness drops to h2, and the growth of instability continues until the minimum film thickness remains greater than hl. For h < hl, &, becomes positive, so that film may eventually assume a steady, finite amplitude, corrugated profile in the form of microdrops separated by relatively flat thinner films. Another possibility is that a small fingerlike projection of the interface has enough pent-up kinetic energy to break through the barrier which exista for do < h < hl,leading to true rupture. While the growth of initial instability is accurately predicted by the linear theory, the questions related to the eventual fate of growing disturbances and the f i a l morphology of the system can only be studied by a nonlinear dynamical formulation. However, some general characteristics of the film instability are already apparent from the linear analysis presented here. In order to make the relationships between the macroscopic and microscopic behaviors physically transparent, it is convenient to consider case8 corresponding to S < 0 and S > 0 separately.
Case I. S < 0; sLw < 0 and Se < 0. The second derivative of the free energy is always negative and decreasing with a decrease in the film thickness (type 1 behavior of Figure 2). The only possible outcome of the film instability is rupture of the film. The time of rupture due to infinitesimal initial perturbations is given by expression 37. Examples include filmsof polar liquids on nonwettable apolar substrates. If the bounding fluid is gas, and the LW component of surface tension of the substrate is lower than the LW component of surface tension of the film liquid, SLwis negative (eq 8b). For films of water on Teflon, SLw=: -5 mN/m (assuming rlLW for Teflon to be 17 mN/m), and Sp= -2ysP = -102 mN/m (y3P for water is 51 mN). On the basis of eq 26, it is easily verified that destabilization due to polar interactions becomes dominant for film thickness less than about 8 nm in this case. The rupture of water films on Teflon due to polar forces is then predicted to be rather instantaneous. Case 11. S 1 0; sLw1 0, Se 1 0. In this case, both the apolar and polar interactions encourage spreading (type 2 behavior in Figure 2) and the contact angle is zero for macroscopic drops. The film also remains unconditionally stable. Examples include films of water on intensely hydrophilic, polar substrates (Sp> 0),24127*29*30 having rlLW larger than 73Lw (21.8 mN/m for water). Case 111. S > 0; sLw < 0 and Se > 0. The above conditions also imply that Sp > ISLw(.A complete spreading of macroscopic drops is predicted from the Young equation. However, thin films become unstable when the condition
61SLW)(dol/h,2)2 > Spexp[(do- ho)/ll
(40)
is satisfied (from eq 22). This case corresponds to type 2 behavior for r#Jh in Figure 2, and the above condition for instability is satisfied for h > h2 and for h < hl (see Figure 5 for estimates of hl and h2). Clearly, for h < hl, rupture is the only possibility,whereas for h > ha, growth of initial instability may lead either to rupture or to a film of uneven thickness. In particular, possibility of rupture exista whenever the condition for instability is also satisfied at h = do, i.e. 61SLWJ(l/do)2 > Sp (41) while Sp> lSLwl,rupture may indeed occur since 611/doI2 >> 1. The polar interactions are not sufficient for film stabilization in such a case. Examples are films of largely monopolar liquids with high yLw (e.g., DMSO)on polar substrates of lower yLW (e.g., Talc, hydrated HSA, and 1g~1.24-34 Case IV. S > 0; sLw > 0, Se < 0. As in case 111, macroscopic drops completely wet the substrate. The above conditions imply that (-Sp/SLw) < 1 or SLw > ISPI. Further if (-Sp/SLw)I0.1 (Figure 51, I$h curve is of type 3 in Figure 2, and instability is not possible. For 0.1 I (-Sp/SLw) < 1, &, curve is of type 4, and the film is unstable in the range of thicknesses hl < h < h2. However, the growth of initial instability tends to saturate as the minimum thickness of the film at localized spots drops below hl. Clearly, the condition for instability in this case JSplexp[(do- ho)/ll > 6(d0l/h,2l2SLW
(42)
when evaluated at h = do gives JSpI> 6(l/dol2SLW (43) The above condition cannot be satisfied since SLw> ISPI. Thus, disintegration of the film in the form of
Sharma
868 Langmuir, Vol. 9,No.3, 1993
microdropswith interveningareas of bare support appears to be improbable. The film interface would, however, assume a steady, spatially nonhomogeneous profile if the mean film thickness is constrained to be between hi and h2, i.e., the liquid loading per unit substrate area is fixed. An example is thin water films on a mucus covered cornea (SLW = 15.5, SAB > -15.5; in mJ/m2)30that remain stable even after wiping and prolonged drying. Case V. S < 0; sLw< 0, P > 0. The macroscopic drop for such a system displaysfinite contact angles. The above > Sp,(-Sp/SLw) < 1, and the conditions imply that JSLwI thin film is unstable whenever 61SLWl(d,Z/h,2)2 > Spexp[(d, - h,)/ll
(44)
is satisfied. Further, when (-SP/SLw) I0.1, the film is always unstable regardless of ita thickness (see Figure 5, and the corresponding type 1 curve for dl)in Figure 2). For 0.1 I(-sp/sLw) < 1, (#Jh is of type 2 in Figure 2, i.e., the film is unstable for h > h2 and alsofor h < hl. The condition for instability at the point of rupture, h = do gives 61SLWl(l/d,)2 > Sp
(45) while this condition is formally the same as in case I11 (S > 0; S L W < 0, S P > 0)discussed earlier,there is an important
difference. Unlike case 111,lSLwI > Sphere,so that rupture of the film is much more likely compared to case 111, as inequality 45 is automatically satisfied. Case VI. S< 0; sLw> 0, P < 0. The above conditions imply 1Sp1> SLw, and behavior of '$h corresponds to the type 4 curve of Figure 2. The film is unstable when the liquid loading is such that the mean thickness falls in the range hl to hp (Figure 5). The condition for instability
lSplexp[(d, - ho)/ll > 6(d0l/h,-J2SLW (46) when evaluated at the minimum thickness corresponding to rupture gives lSpl> 6(Z/d0)'SLW (47) Although (SpI> SLwin this case, 6(l/d0)~ may be a large quantity, so that inequality seems difficult to satisfy. A true breakup of the film, while not impossible,seems not likely for most systems. Thus, the contact angle of a thin film may remain zero (wetting film) even when a finite contact angle is predicted for macroscopic drops of this system (S < 0). Water films (bounded by air) on most polar substrates fall in this category. Examples of such substrates include PMMA, hectorite, corneal epithelium, HSA, cellulose acetate, poly(viny1 chloride), polystyrene, et^.^"^^ Finally, it may be noted that whenever macroscopic drops display finite contact angles (S< 0), conditions for the film stability/instability may also be interpreted in terms of 8, by replacing Spfrom Young's equation.
sp= COS e - 1) - sLw
(48)
Thus, all conditions of the film stability/instabilitymay also be obtained in (SLW, 8) parameter space. As is pointed out in the introduction, the above concepts of the film stability, and its relationship to wetting by macroscopic drops, are important for heterogeneous nucleation, formation of "dry" spots in macroscopic films, and equilibrium film pressure,4145 among other applications. (41) Hu, P.; Adamson, A. W. J. Colloid Interface Sci. 1977,59, 605. (42) Tamai, Y.;Mataunaga, T.; Horiuchi, K. J.Colloid Interface Sci. 1977, 60, 112. (43) Whalen, J. W.; Hu, P. C. J . Colloid Interface Sci. 1978,65,460.
In the classical theory of nucleation, the critical size of the nucleus and the energy barrier for its growth depend on its free energy, morphology, and wetting properties (contact angle), all of which may be quite different from their macroscopic counterparts. In the context of dry spot formation in films employed for heat and mass transfer, the following statements are typical of the current thinking in the literature: "It is interesting to note in retrospect that while the value of this contact angle plays no role in determining the initial stability of the film to dry patch formation, it is the dominant factor in their removal ...";22 whenever the thickness of the film is of the order of a few hundred angstroms or less, it will spontaneously thin itself and rupture, leavingadry patch."22 ".... in some still unknown way, the contact angle influences the ease of formation of dryspot."12 The first two statements are incorrect in that not all thin films necessarily rupture, and when they do, the instability is indeed related to macroscopic parameters of wetting (e.g., contact angle and spreading coefficients). As regards the last statement, the theory presented here provides the missing link between the film stability and the contact angle. Finally, it is clear that a substantial equilibrium f i i pressure, a, can exist only if a stable (wetting) thin f i i , which is contiguous with the drop, can form on the substrate. Morphology of the thin film is important, as disjointed microdrops on the substrate cannot engender the film pressure for the macrodrop,4l regardless of the amount adsorbed. Thus, while an effective (or fictitious) film thickness can be easily determined from the measurementa of amounbad~orbed,4~?~~ these amounts would not necessarily correlatewith the spreading (film)pressure if the film is unstable. For example, water on low-energy apolar surfaces (e.g., Teflon and polyethylene) does not show appreciable film pressure,u as a stable continuous thin film cannot exist for such systems, unless residual polar sites are present. Examples of apolarsystems include hydrocarbon films (3), bounded by a low-energy apolar substrate (1) (e.g., Teflon, ylLwr 17 mN/m)26and a polar liquid (2) with high LW component of surface tension (e.g., glycerol; ypLw= 34 mN/m).26 Clearly, Spis zero and whenever the surface tension of the hydrocarbon film satisfies the condition ylLw < yzLW< ysLW,SLwis positive (eq 8b). The hydrocarbon f i i should,therefore,be stable. In view of this, a finite film pressure, r e , should be included in the Young equation for interpretation of contact angles of water drops in hexane on Teflon and also for hexane drops in water on Teflon.42145In the former case, a hydrocarbon film is expected between the water drop and substrate, whereas in the latter case, the film surrounds the hydrocarbon drop. The sign of r e is, therefore, different in these two cases. The logic behind correlation of a, with the film stability is general and is easily extended to different types of polar systems based on the film stability results obtained here. As a final example,consider water filmson intenselypolar substrates (e.g., dry human serum albumin, HSA, for which, ylLw = 41, y1+ = 0, 71-= 20, and S L W = +16.2, Sm = -56.4; all in mJ/m2).s This correspondsto case VI discussedearlier. The water films of thicknesses less than hl (about 0.4 nm in Figure 5) are certainly stable, but even thicker films are also unlikely to rupture completely. A significant film pressure and contact angle hysterisis may then be anticipated for water drops on hydrated HSA. In general,based
"...
(44) Fowkes,F. M.;McCarthy, D. C.;Mostafa,M. A.J. Colloidlnterfuce Sci. 1980, 78, 200. (45) Janczuk, B.; Chibowski,E. J. Colloid Interface Sci. 1983,95,268.
Stability of Thin Fluid Films
Langmuir, Vol. 9, No. 3, 1993 869
on Figure 2, the film pressure should be largest for type 3 systems followed by type 4 and type 2 systems.
Summary Important qualitative conclusions may be summarized as follows: 1. Stability (wetting) behavior of thin films in completely apolar (Sp 0) and completely polar (SLW 0) systems parallels the degree of wetting by corresponding macroscopic drops. The instability becomes stronger as the contact angle increases. The above conclusions are also valid when Spand SLwboth have the same sign. 2. When SLwand Spare of opposite signs, there is no unique correlation between the wetting behavior of macroscopic drops and stability of thin f i i s . Conditions are analyzed under which a thin f i i is stable (wetting) despite a fiiite macroscopic contact angle, and also
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conditions that ensure instability even when macroscopic drop shows completewetting. The film instability, when it occurs, is confined to a range of film thicknesses. There are two possible outcomes of the instability: (a) true rupture of the film leadingto microdrops with f i i t e contact angle with the support; (b) saturation of instability resulting in microdrops that are in equilibrium with thin films of nonuniform curvature. On the basis of characteristics of the second derivative of the free energy, and linear stability analysis, simple criteria are derived for the above scenarios. The criteria involve macroscopic parameters of wetting, e.g., spreading coefficients, contact angle, and f i i interfacial tension. However, detailed characterization of microscopic shapes and contact angles of microdrops requires a fully nonlinear dynamicaltheory, which is being pursued currently.
