J. CHEM. SOC. FARADAY TRANS., 1994, 90(4), 625-627 625 Stability of Thin Polar Films on Non-wettable Substrates Ashutosh Sharma" and Ahmad T. Jameel Department of Chemical Engineering, Indian Institute of Technology at Kanpur, Kanpur 208 016, India The linear and non-linear stabilities of thin (40 nm) apolar and polar films (water, polymers) on non-wettable substrates are correlated to the macroscopic parameters of wetting. The wavelength of instability and the time of film rupture decline as the substrate becomes less wettable by macroscopic drops. The instability of relatively thick (~12nm) completely polar films (eg. water films bounded by octane) evolves very slowly, and the sub-strate may appear to be wettable, notwithstanding a large equilibrium contact angle.In contrast to the apolar films, the non-linear interactions in polar films may remain significant even for the small-amplitude thermal perturbations. Thin (<50 nm) fluid films on non-wettable surfaces are inher- ently unstable'-6 and the growth of interfacial deformations leads to the film breakup and retraction, resulting in drops of finite contact angle. The linear,'-3 as well as the non-linear4*' aspects of the instability have been extensively studied for apolar films (e.g. hydrocarbons) where the excess intermolec- ular energy is derived solely from the apolar Lifshitz-van der Waals (LW) interaction^.^.' However, polar liquids (e.g. water) on the apolar and polar substrates also experience polar interactions variously described as the hydrophobic attraction, hydrogen bonding, 'acid-base' interactions, et~.~.' It is well known' that the interfacial (surface) tensions and the equilibrium wettability (contact angle) of water on any surface can only be described by inclusion of polar 'acid- base' interactions.Indeed, the equilibrium contact angle (wettability) of water, 0, is related to the apolar (nSvLw)and the polar (nSvp)components of the total spreading pressure (ns)by the Young equation cos 0 = 1 + -; ns = nSJ-W + nS,P (1)723 where y23 is the film interfacial tension against the bounding fluid. The term interfacial tension (y,) is reserved for the interface between two condensed media i and j. Ifj is a gas, the term surface tension (yi) is more appropriate for denoting the specific surface energy of material i.nS.' and nSvLware defined in terms of the LW and polar components of various surface (yi) and the interfacial (yij) tensions as' (1 = substrate, 3 = film, 2 = bounding fluid) (24 nS*' = yy2 -y!3 -yp23 = -2y; (if 1 and 2 are apolar) (2b) Thus LW forces promote the non-wettability (film instability) for ns*Lw< 0, which is the case for water films (bounded by air) on low-energy surfaces (e.g. Teflon, aminosilane-treated glass fibre, etc.) with ykW < yiw ( =21.8 mJ m-2 for water).' nSvLwis also negative for water films sandwiched between higher-energy media, i.e. y;", yk" > yf;" also implies nSvLw< 0 from eqn. (2a). Examples are water films on most polymeric and biological substrates bounded by hexadecane.The polar forces also engender non-wettability as nsVpof water on almost any surface is negative due to the large polar cohesive energy of water molecules (75 = 51 mJ m-2).8 The minimum nS*' = -102 mJ m-2 for water is obtained on the apolar substrates (e.g. Teflon), but increased polar inter-actions with the substrate make it less negative [eqn. (2b)l. ns*Lwand nSvpcan be easily evaluated from measurements of contact angles of an apolar liquid and water, in conjunc- tion with eqn. (l).398Briefly, 74" for the substrate can be evaluated with the help of eqn. (1) and (2a) by measuring the contact angle of an apolar hydrocarbon liquid (nS*' = 0) of known surface tension. nSsLwfor water on the same substrate is now obtained from definition (24.Finally, ns*pfor water is determined from eqn. (1) by directly measuring the contact angle on the substrate. In what follows, we investigate the influence of nSvp(and the contact angle) on the stability and kinetics of rupture of water films. Theory The following leading-order non-linear equation describes the evolution of the film thickness, H(x, t)4*5 (3) where t, x and p are time, lateral space coordinate (parallel to the film surface) and the film viscosity, respectively. 4 is the excess energy of the film per unit volume due to intermolecu- lar interactions, which is related to the Gibbs energy per unit area, AG by 4 = aAG/dH, where AG is given by3*798 AG(H) = nSvLW(d;/H2)+ nsyp exp[(do -H)/q (4) where do is the 'cut-off equilibrium distance (=0.158 nm)' where the Born repulsion takes over, and 1 is a correlation length for polar 'acid-base' interactions (I z 0.2-1 nm for water; the best estimate from one study is 0.6 nm).' Clearly, ns,Lwis related to the effective Hamaker constant by the defi- nition, A = -12ndinS*LW,3*8and the change of the Gibbs energy, AG = G(do) -G(m) is as expected given by 2s.Lw + nS,' = ns = y12 -yI3 -y23.Clearly, it is advantageous to write the Gibbs energy of the thin film in the form of eqn. (4), which only involves the macroscopic parameters (nSvLwand nS*') of wetting. As is discussed in the introduction, ns*Lwand nS.' are easily obtained by measurements of equilibrium contact angles of macroscopic drops.In contrast, a direct estimation of the effective Hamaker constant and the strength of polar interactions poses many dificulties, especially for the ill-defined, modified surfaces often encountered in practice. Results and Discussion The linearization of eqn. (3) and (4) around the mean film thickness, h, gives the fastest growing mode of instability of initial amplitude, E, H = h + E sin(2nx//2,)exp(ot), from which a (linear) estimate of the time of rupture' is obtained by setting H = 0, i.e. t, = ln(h/e)/o. The fastest growing (dominant) mode of the linear theory evolves on a length scale (wavelength), A, A: = -(4Z2y23h4/3dz ,s*Lw)( 1 + (h2/doI)2xs9p x exp[(do -h)/fl/6~~.~~} (5)-The linear theory estimate for the minimum time of rupture is t, = by23 h'/3(~~~)~d~](l+ (h2/doO2xp x exp[(do -h)/fl/6xLw}-2 ln(h/&) (6) Clearly, decreased wettability (more negative xLwand x? encourages faster rupture, caused by shorter waves.Results (5) and (6)can also be interpreted in terms of 8 by elimination of xpfrom these equations and eqn. (1). Fig. 1 illustrates the influences of mean film thickness, and the polar spreading pressure (contact angle) on the (linear) time of rupture for a fixed value of xLw= -15 mJ m-2 (water on perfluorolauric acid substrate). For films thinner than 10 nm, polar interaction is significant and a decreased wettability due to the polar interaction (more negative x? causes a greater destabilization of the film.However, for thicker films, a more rapid (exponential) decay of polar forces renders them ineffective and the time of rupture is then con- trolled by the apolar LW forces alone [see eqn. (5) and (6)]. An interesting example in this context is a water film on any substrate bounded by octane, for which ykw = ybw = 21 mJ m-2, and xLw= 0 from eqn. (24. Such completely polar films do not experience excess LW interactions, and relatively thick films should therefore appear to be rather stable, not- withstanding a large (negative) xp and a large equilibrium contact angle, from eqn. (1). The time of rupture (time for appearance of a true three-phase contact zone) for completely polar films is obtained from eqn. (6)by setting xLwto zero, uiz.t, = (%'23/h){l2 exp[(h -d~)/fl/x'h}~In(h/&) (7) For water films bounded by octane, the maximum possible value of I xpI is 102 mJ m-2, from eqn. (2b),and therefore, the minimum time of rupture for water films thicker than 12 nm is more than 30 min. Thus, the contact angle hysteresis in these cases may also be of kinetic origin, which is due to the slow dynamics of the formation of an ultrathin contact zone J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 where the polar interactions take over. In conclusion, for comparable values of the spreading pressure (equilibrium contact angle) relatively thick (>8 nm) films of completely polar systems are considerably more stable than the apolar films. Once a polar film is formed, the surface may remain perfectly wettable for a long time, even though the final equi- librium contact angle, from eqn.(1) is large. However, such thick polar films can still recede and evolve more rapidly into equilibrium drops if local film thickness is decreased (66nm) by external means so that the polar interactions can cause destabilization (receding motion) and a growing microscopic contact zone can form. While the qualitative aspects of the growth of instability are well described by the linear stability [eqn. (5) and (6)],the quantitative predictions require a direct numerical solution of the non-linear equation of evolution (3). The non-linear eqn. (3) was solved by a pseudo-spectral method: Fourier collo- cati~n,~~'~together with periodicity conditions over the wavelength, x E (0,A) and a small amplitude initial dis- turbance, H(x, 0) = h(1 + 0.01 sin 2xx/A).A cosine dis-turbance merely phase-shifts the periodic profile of the film. 24 collocation (grid) points were found satisfactory, and the resulting set of 24 non-linear coupled ordinary differential equations were integrated in time by GEAR algorithm for stiff equations. The length scale (A,,) of the fastest-growing wave, and the corresponding minimum time of rupture t, were determined by solving eqn. (3) for many different values of A in the neighbourhood of AL . The ratios of the non-linear and the linear predictions are shown in Fig. 2. When the polar interactions are important [h < 8 nm, curve (a)], the instability evolves on a length scale (unit cell size), A,,, that may be large compared to the linear estimate, A,.However, for thicker films controlled largely by LW forces, A,, x A,, at least for small initial amplitudes (e.g. thermal perturbations). The size of the unit cell (ca. A,') determines the number of holes per unit area. Experiments with polymer films heated above the glass-transition temperature prove that the equi- librium morphology of the ruptured film is also governed by this factorq6 Further, Fig. 2(b)shows that the non-linearities of both the apolar and polar interactions accelerate the film breakup, i.e., t, < t,. Finally, curve (c) shows the ratio of non-linear times of rupture as obtained for A,, and A,, respectively.For small initial amplitude, t,(A,) x t,(A,), which obviates the need for a 0.2Lit 0 4 a 12 16 0 5 10 15 20 25 h/nm hlnm Fig. 1 Influence of film thickness, nS*' and 8 on the normalized time Fig. 2 Comparison between non-linear and linear theories: (a) of rupture from the linear theory (2vLW = -15 mJ m-'). nssP= (a) 0, A,,/AL; (b)tJtL; (c) tn(An)/tn(AL). (7?s,Lw = -17.17 mJ m-2, nSvp= -60 (b) -5, (c) -10, (6)-30, (e) -60 and (f)-102 mJ m-2. 8 = (a) 37, mJ m-2; symbols represent computed values, lines are only to guide (b)44, (c) 49, (d)68, (e)92 and (f)127 degrees. the eye.) J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 repeated solution of eqn. (3) for different A in order to deter- mine the minimum time of rupture. As is the case for the apolar however, the results may be different for rela- tively large amplitude mechanical perturbations, where non- linear effects are significant from the beginning (work under progress).The theory provides, for the first time, a correlation between the thin-film stability and the macroscopic param- eters of wetting, which should be useful for the design and interpretation of future experiments involving thin polar films (e.g. water, polymers6). References 1 A. Sheludko, Adu. CoIIoid Inteqace Sci., 1967, 1, 391. 2 E. Ruckenstein and R. K. Jain, J. Chem. SOC.,Faraday Trans. 2, 1974,70, 132. 3 A. Sharma, Langmuir, 1993,9, 861, and references therein. 4 M. B. Williams and S. H. Davis, J. Colloid Interfuce Sci., 1982, 90,220. 5 A. Sharma and E. Ruckenstein, J. Colloid Interface Sci., 1986, 113,456. 6 G. Reiter, Phys. Rev. Lett., 1992,68, 75; Langmuir, 1993,9, 1344. 7 J. H. Israelachvili, Intermolecular and Surface Forces, Academic Press, New York, 1985. 8 C. J. van Oss, J. Dispersion Sci. Technol., 1991, 12, 210; C. J. van Oss, M. K. Chaudhury and R. J. Good, Chem. Rev., 1988, 88, 927. 9 C. Canuto, M. Y.Hussaini, A. Quarteroni and T. A. Zang, Spec-tral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988. 10 A. Sharma and A. T. Jameel, Colloid Interface Sci., 1993, 161, 190. Paper 3/06794G ;Received 9th September, 1993