Investigation of Equilibrium Wetting Films of n-Alkanes on a-Alumina By TERENCE D. BLAKE Research Division, Kodak Limited, Headstone Drive, Harrow, Middlesex HA1 4TY Received 3rd April, 1974 A review is given of previous disjoining pressure investigations of liquid films on solids, and the advantages of combining such investigations with corresponding vapour adsorption studies are discussed. The relevant thermodynamic framework is presented in outline. Direct measurements of the disjoining pressure of stable wetting films of n-octane and n-decane on a-alumina are reported for films of from 20-80 nm in thickness. The results obtained, together with data for much thinner films derived from a previous gravimetric study of the vapour adsorption of n-decane on the oxidised surface of aluminium foil, are compared with the predictions of Lifshitz and London-Hamaker theories of dispersion forces.It is found that both theories correctly predict the disjoining pressure and differential free energy of formation of the films, provided the films exceed monolayer coverage, and provided due allowance is made for retardation if the films exceed a thickness of about 5 nm. In making these predictions, it is sufficient to assume that the films have molar volumes, molar entropies and dielectric properties identical with those of the parent bulk liquids. This paper reports a disjoining pressure study of wetting films of n-alkanes on cc-alumina and compares the results with the predictions of Lifshitz and London- Hamaker theories of dispersion forces.Much of the previous work on systems of this type has centred on measurements of vapour adsorption, and there has been a tendency to neglect the disjoining pressure approach. This paper seeks to partly redress the balance and to suggest, through example, that the two approaches are profitably treated as complementary. The paper contains a certain amount of review material and a section is devoted to an outline of an appropriate thermo- dynamic framework. showed that when a gas bubble is pressed against a hydrophilic solid surfaye (such as a plate of mica or glass) immersed in water, the intervening liquid slowly thins to leave a uniform equilibrium film of considerable thickness (fig. 1). The film behaves as though there were an excess pressure, II, acting normal to the film and opposing further reductions in film 'thickness, 1.This excess pressure, originally termed by Derjaguin the " wedging apart " or " disjoining " pressure, represents the difference between the pressure within the bubble, pb, and that in the bulk liquid adjacent to the solid surface,^' : i.e. The original investigations by Derjaguin and co-workers n = y b - p Z . (1) From elementary considerations of mechanical equilibrium, it follows that for stable films, I I > O (2) and (anpl) < 0. 192 (3)T. D . BLAKE 193 If a film is stable at all thicknesses, then the liquid is said to completely wet the solid. Thus a condition for complete wetting is that the function n = II(2) decreases mono- tonically from some positive value at I = 0 and approaches zero as I -+ co.A more general condition for complete wetting is given by; where 07, asz and 0: are the equilibrium tensions of the solid/vapour, solid/liquid and liquid/vapour interfaces respectively, and the subscript indicates saturation vapour pressure. The significance of this expression has been discussed in great detail by Melr~se.~ Discussions of wetting that involve the concept of disjoining pressure have been given by Frumkin and by Derjaguin and Shcherbak~v.~ aso' = cTsz+a;, (4) liquid - 5 Y - L = p b - p l FIG. 1.-Formation of a stable wetting film between a bubble and a plate. The phenomenon of disjoining pressure is believed to result from three different types of solid/liquid interaction : compression of electrostatic double layers within the film; van der Waals forces of attraction between liquid and solid-chiefly London dispersion forces; and forces of a third kind,6 usually assumed to be entropic in character, such as the build-up of solvation layers by hydrogen bonding.Whilst the first two types of interaction have a firm experimental foundation, the significance of the third type is still contr~versial.~ In view of the material importance of solid/liquid interactions, it is surprising that disjoining pressure studies have been so little exploited until quite recently. The bubble-against-plate (b.a.p.) technique is a considerably more direct way of investi- gating solid /liquid interactions than some of the alternative approaches more com- monly employed, such as classical studies of the coagulation of lyophobic colloids.' Presumably, the slow progress has been due to the exceptional experimental difficulties encountered, including non-reproducibility and inhomogeneity of solid surface^,^ greasy contamination, foreign particles, and vibration.l0.Thus, for example, Elton l2 and Evans l 3 were unable to repeat the Russian results and respectively attri- buted them to electroviscous effects and contamination. 1 - 7194 n-ALKANE FILMS ON a-ALUMINA More recently, however, Kitchener and co-workers lo* 0 l4 successfully applied the method to show that thick ( I > 20 nm) aqueous films on silica are stabilised principally by electrostatic forces, and are not significantly influenced by forces of a third kind. Detailed studies by Schulze and Ciclios l 5 lead to essentially the same conclusions, although the thinnest films reported (10 to 15 nm, at high ionic strengths) may have been stabilised by dispersion forces.In addition, Tabor and Roberts 16* l7 have shown that electrostatic double layer theory accounts reasonably well for the behaviour of aqueous films at disjoining pressures as high as 6 x lo4 N m-2. In these experiments, which involved a glass substrate and a transparent rubber ball in place of a bubble, dispersion forces, considered alone, probably produced a nett attraction between the materials on either side of the film and hence a negative contribution to the total disjoining pressure. Calculations l7 indicate that this contribution is about - 4 x lo2 N m-2, i.e. about two orders of magnitude less than the total pressure.In the new experimental work presented here, the b.a.p. technique has been used to investigate a system in which dispersion forces alone are responsible for the forma- tion of stable wetting films. A simple system was chosen so that results could be compared unambiguously with theory. Synthetic a-alumina was selected as the solid substrate because of its simple dispersion properties (see below), high refractive index and therefore high dispersion interaction ; it had the additional advantages of being rigid, chemically inert and crystallographically well-defined. n-Octane and n-decane were selected as the liquids because they are non-polar and have low refrac- tive indices. Preliminary calculations indicated that even with such a favourable system, films of substantial thickness would exist only at very low disjoining pressures.Measurements were therefore made over the range 0.5 to 50 N m-2. Previous disjoining pressure studies of well-characterised systems of this type are few. However, Scheludko and Platikanov l8 reported a 24 nm thick equilibrium film of benzene on mercury at II E 70 N m-2, and Schulze l9 reported a 28 nm equilibrium film of n-octane on silica at 16.2 N m-2. This last value is in good agreement with dispersion force calculations.20 The formation of wetting films on solids has, of course, been investigated many times before by conventional vapour adsorption techniques. However, because of capillary condensation, these investigations have usually been restricted to films of a few monolayers or less.Where this difficulty has been overcome (as, for example, in the ellipsometric study of adsorption on silica by Derjaguin and Zorin 21), then the very precise temperature control necessary at very high relative pressures has still restricted equilibrium measurements to comparatively thin films. Recent investiga- tions 22* 23 of the adsorption of liquid helium on alkaline-earth metal fluorides were exceptional. Because of the unusual properties of liquid helium, it was possible to use an acoustic phonon spectrometer 24 to investigate helium films of from 1 to 20 nm in thickness. The results obtained were shown to be in excellent agreement with dispersion force calculations. 25 With more conventional systems, it would appear that the most profitable course is to treat vapour adsorption studies and disjoining pressure measurements by the b.a.p.technique as complementary. For, whilst adsorption measurements are usually restricted to films of less than, say, 10 nm in thickness, the b.a.p. technique is most conveniently applied to much thicker films. This dual approach is exemplified in the discussion section of this paper where results, both from the present investigation and from a previous vapour adsorption study, are compared with the results of dispersion force calculations. For the purpose of this comparison it is necessary to employ a thermodynamic framework appropriate to both b.a.p. and adsorption(b) PLATE 1.-(u) Appearance of a typical film when viewed from beneath by reflected monochromatic light.(6) Type of interference pattern obtained when the bubbles are positioned just sufficiently above the alumina surface for no flattening to occur. [To face page 194T. D. BLAKE 195 systems. This framework, which involves certain assumptions about the nature of films, is therefore outlined in the following section. Only planar wetting films of a single component are considered. A general account of the thermodynamics of films has been given by Rusanov.26 THERMODYNAMIC FRAMEWORK According to the classical view of interface^,^^ whilst the coniponent of the pressure tensor normal to a plane interface is equal to the pressure in the contiguous bulk phases, the tangential component takes a different value which varies with position along the normal.It is this anisotropy of the stress tensor that gives rise to the phenomena of surface tension and surface pressure. In the case of a film in equili- brium with its parent bulk liquid (as in the b.a.p. system), the mechanical anisotropy is manifest not only in the form of a film tension, but also as an apparent excess pressure acting normal to the film, i.e. the disjoining pressure. Nevertheless, the mechanical properties of a plane film, like those of a plane interface, may be adequately described in terms of a single film tension and a single isotropic pressure, pf, equal to the normal pressure exerted by the contiguous bulk phases.26 For a film on a solid, the film tension may be identified with 0'. For a film in the b.a.p. system, pf = pb, whereas for an adsorbed film in equilibrium with its pure vapour at pressure p", pf = p'.The magnitudes of the extensive thermodynamic properties of a film depend upon the choice of the positions of the two principal dividing surfaces that constitute its boundaries. Although in principle this choice is quite arbitrary, in practice it will be operational. That is to say, it will be determined by the nature of the experimental method and the system. For present purposes, it is convenient to assume that the density of the solid remains constant up to a dividing surface chosen so that all the molecules of the solid component are in the solid and none are in the film. For a rigid, inert solid such as u-alumina, this is a realistic choice. The position of the second dividing surface relative to the first is fixed by either ascribing a value to the molar volume of the film component, or by employing the experimentally determined film thickness.Provided one is not concerned with the detailed structure of a film, it may be treated as a mechanically anisotropic but otherwise homogeneous phase. Thus, by making use of Gibbs' thermodynamic treatment of bulk phases,28 the partial molar Helmholtz free energy of a single-component film can be expressed as : where Uf, Sf, V', Af and nf are respectively the internal energy, entropy, volume, area and number of moles of the film at chemical potential pf and temperature T. Simi- larly, for the parent bulk liquid at the same temperature,196 I1-ALKANE FILMS ON a-ALUMINA whence, on subtracting (6) from (5), one obtains the differential Helmholtz free energy of film formation : This expression gives the coefficients for variations in free energy resulting from iso- thermal transfer of molecules between film and bulk liquid at specified pressures.The expression is equally applicable to both experimental systems under consideration. Consider the b.a.p. system. If the liquid is assumed to be incompressible so that film and bulk liquid have the same molar volume, i.e. ($)T,Af = r$)T = d, then, since the film is in equilibrium with its parent bulk liquid, pf = p' and (7) reduces to ATf = - d(pf - p'). (8) IT = -Affiu! (9) But, pf = pb, hence pf -p' is equal to the disjoining pressure, n, and Now consider the adsorption system. The film is in equilibrium with its vapour, which may be assumed to be a perfect gas.Moreover, away from the critical point where p: is the saturation vapour pressure of the liquid. (7) reduces to : Hence, for this system, ATf = RT In pvIp& (10) Eqn (7), (9) and (10) provide the thermodynamic relationships required for the discussion section of this paper. EXPERIMENTAL METHOD The techniques used in this study were similar to those described previously." However, they were employed with greater precision. The apparatus, which was mounted on an antivibration table of low periodicity, is shown schematically in fig. 2. A bubble of purified nitrogen held in an inverted cup (D) was lowered slowly with a micromanipulator (A) to press lightly against an optically-polished, flat, a-alumina disk (F) immersed in the test liquid (E).Films formed between bubble and disk were observed from beneath by reflected monochromatic light (wavelength A = 435 or 547 nm), using a low-powered microscope (x 100 or x 200 magnification). The appearance of a typical film together with its surround- ing " Newton's rings " interference pattern is shown in plate l(a). The disjoining pressure was varied over the required range by varying the size of the bubble, using bubble holders of from 2 to 25 mm diameter. Most experiments were carried out at 294& 1 K.T . D . BLAKE 197 Q FIG. 2.-Schematic representation of the disjoining pressure apparatus : A, micromanipulator ; B, Perspex resin enclosure ; C, fused silica cell ; D, bubble holder ; E, n-alkane ; F, optically-polished a-alumina disc ; G, optical window; H, x 10 microscope objective; I, interference ater (435 or 547 nm) ; J, heat filter ; K, condenser ; L, 250 W, high-pressure mercury lamp ; My silica plate ; N, collimator ; 0, beam-splitter ; P, prism ; Q, 35 mm camera ; R, eyepiece.Disjoining pressures were determined from photographs of the interference pattern obtained when the bubbles were positioned just sufficiently above the alumina surface for no flattening to occur [plate l(b)]. Since the base of each bubble was then essentially spherical, the pressure drop across its surface was given simply by Ap = 2azv/rb. The radius of curvature, rb, was calculated from the expression where ri and ri are the radii of the ith and jth interference fringes, and R! is the refractive index of the liquid.On pressing the bubble against the alumina surface, providing the radius of the film so formed was much less than rb, the pressure in the bubble was negligibly altered and Ap was equal to II ; in practice, film radii were always less than rb/lO. The thickness of each film was determined from its reflectivity 29 I (4sf)2 + (4fb)2 + 24sf4fb cos 6 = I, = 1 +(#sf4fb)2 + 2#sf4fb cos 6 ' where Pf = ( & - H ~ ) / ( N ~ + + ~ ~ ) , fifb = ( ~ ~ - + w ~ ) / ( ~ ~ + & b ) and 6 = 47c~~Z/A. I. and I are the intensities of incident and reflected light, and &, wzf and are the refractive indices of solid, film and bubble respectively. The refractive index of the film was assumed to be the same as that of the bulk liquid. Fig. 3 shows p as a function of I for films of n-decane on a-alumina at the two wavelengths used.The interference order was established by observing the shift in p on changing from one wavelength to the other. To measure p, fine-grain photographs of the films and their surrounding interference patterns were taken using a 35 mm camera attachment. The negatives were developed under standard conditions and scanned with a calibrated Joyce-Loebel double-beam micro- densitometer to give density profiles similar to that shown in fig. 4. The reflectivity of each198 ll-ALKANE FILMS ON CI-ALUMINA film was found by comparing its image density, d, with dmax and dmin, the densities of the maxima and minima, Pmax and pmin, given by eqn (11). Over the small density range involved, (P max + Pel (P + P,) ’ dmax- d = log and The term pe, which accounts for any extraneous light entering the microscope, is easily eliminated from these equations to give p.Use of the appropriate p against I curve (fig. 3) then gives I directly. 1 I I I I I 5 0 100 1 5 0 200 2 5 0 300 V Ilnm FIG. 3.-Variation of reflectivity, p, with film thickness, Z, for films of n-decane on a-alumina; curves calculated using eqn (11) for experimental wavelengths, A, of 435 and 547 nm. t d (direction of scan FIG. 4.-Image density profile of a film and its surrounding Newton’s rings interference pattern. This profile corresponds to a film thickness of 46 _+ 4 nm. Because of fluctuations in lo, it was necessary to determine d, dm, and &in from the same photograph. However, as can be seen from fig.4, the contrast between succeeding maxima and minima gradually decayed as one scanned away from the fdm, with even the first minimum being slightly affected. This attenuation was due to the increasing curvature of the bubble. Nevertheless, examination of the density profiles of very thin films and slowly thinning films exhibiting broad central maxima or minima, showed that sufficiently accurate values of dmax and &in could be obtained by linear extrapolation of the attenuated valuesT . D . BLAKE 199 The microdensitometer was calibrated for the chosen combination of wavelength, photographic film and development by first exposing a sample of the film to light of the appropriate wavelength through a linear density wedge of known slope. The film was then developed in the given way and scanned with the microdensitometer to give the required calibration curve.MATERIALS AND PREPARATION Synthetic a-alumina disks, 20mm in diameter and 2mm thick, were supplied by Fred Lee and Co. Ltd. They were ground flat and optically polished by the Department of Mining and Mineral Technology, Imperial College, London. Interferometric examination showed that the resulting surfaces were sensibly flat and free of visible irregularities apart from a few fine scratches. Fused silica cells (C, fig. 2) were supplied by Jencons Ltd. with 50-mm diameter optically flat bases. Before use, a cell containing an alumina disk was ultrasonically scoured with Teepol solutions followed by freshly distilled water. The cell was then covered and baked in an oven at about 1000 K for 12 h.After cooling, the cell was immediately filled with the test liquid. The n-octance and n-decane (99 % by g.l.c., as supplied by B.D.H. Ltd.) were exhaustively dried over molecular sieve, percolated through a column of activated alumina to remove any traces of polar impurities and distilled directly into the cells as required. White Spot nitrogen from a compressed gas cylinder was purified over activated charcoal and filtered to remove particles. The gas was inserted into the bubble holders via a micro- pippette. For the remaining materials, the values listed in table 1 were used. The refractive index of nitrogen was taken to be 1.000. TABLE 1 .-REFRACTIVE INDICES 435 1.777 1.406 1.421 547 1.767 1.399 1.414 0 a-alumina is bi-refringent ; values listed are arithmetic means.t/nm a-aluminaaJo n-octane 31 n-decane 31 RESULTS As expected, both n-octane and n-decane gave equilibrium wetting films that persisted, apparently indefinitely, at all disjoining pressures investigated. The films were evidently stable, since equilibrium thickness could be approached from either direction and films were not disrupted even by severe vibration. The experimental data are shown in fig. 5, with ll plotted as a function of I for both liquids. At the highest pressures (> 30 N m-2), equilibrium thickness was effectively attained within 10 min. At the lowest pressures (< 1 N m-2), equilibrium was approached only after 1 to 2 h. Usually, films were monitored for extended periods (more than 15 h in a few cases) after the attainment of equilibrium.The uncertainty associated with measurement of I is difficult to assess. The error bars on each datum point in fig. 5 represent an attempt to estimate the maximum probable error from a number of sources. These include uneven illumination of the film and photographic noise ; however, residual roughness of the alumina surface, and uncertainty in the extrapolated value of dmax are thought to be the main sources of error, especially for I < 30 nm. Larger surface defects and dust particles did not present a problem as these were clearly visible and could be avoided. One of the advantages of the photographic method is that it enables one to select only representa- tive areas of the film for measurement. For films thicker than about 60nm, the disjoining pressures were so low that bubbles became very susceptible to mechanical vibration; hence it was difficult to2,OO n-ALKANE FILMS ON CC-ALUMINA ensure that films were at equilibrium.Measurements of II were also affected by vibration. However, reproducibility was usually better than 2 %. Consequently, the contribution to the overall error was negligible. 2 0 4 0 6 0 8 0 100 FIG. 5.-Dependence of disjoining pressure, lI, on film thickness, I, for films of n-alkanes ona-alumina. Experimental data : 0, n-decane ; a, n-decane after irradiation ; 0, n-octane. Curves indicate theoretical contribution of dispersion forces to TI calculated according to : -, Lifshitz theory ; -- , London-Hamaker theory ; - - - - , London-Hamaker theory, but allowing for retardation.Theoretical curves are shown for n-decane only. DISCUSSION Before comparing the experimental results with the predictions of contemporary theories of dispersion forces, it is useful to consider some of the assumptions under- lying both the theories themselves and their application. There are currently two fundamentally different approaches to the calculation of dispersion forces between condensed media. One of these, the microscopic approach of London 32 and Hamaker,33 is based on the summation of individual, molecular interactions. The other, which is due to Lifshitz and co-w~rkers,~~ treats the inter- acting media as continua and therefore involves only their macroscopic, electro- dynamic properties. Both have been the subject of recent review^.^^-^^ A number of authors have used the microscopic approach to calculate the contribu- tion of dispersion forces to the disjoining pressure of a liquid film on a solid.Schel~dko,~ for example, has integrated London's expression for the energy of interaction between two molecules separated by a distance r, over all pairs of molecules, to obtain U = - B r 6 , (1 3)T . D. BLAKE 20 1 where dsl and dL1 are the Hamaker constants for solid/liquid and liquid/liquid interactions respectively : and dl' = nz(N/v')2B" dsL = T C ~ ( N ~ / U ~ V ~ ) B ~ ~ . Like eqn (9), these expressions contain the simplifying assumption that (a Vf/dnpT,Af = v'. If a corresponding assumption is made about the entropy of the film,g9 39 i.e. (> = (??)T = sl, anf T,Af where s1 is the molar entropy of the liquid, then it follows from (7) and (14) that Hence, from (9) AJf = - ~'(d'' - d1')/6nl '.(1 5 ) II = ( d s ' - d " ) / 6 ~ Z 3 , (16) whilst, from (lo), for an adsorbed layer, In p'/p; = - v z ( d S L - d11)/6n RT 13. Eqn (17) is essentially that proposed by Frenkel,39 Halsey 40 and Hill 41 for the multi- layer adsorption of gases. Because the propagation of electromagnetic radiation is not instantaneous, dispersion forces fall off more rapidly with distance than predicted by eqn (13). The effect, which is known as retardation, has been investigated theoretically by Casimir and Polder.42 They have shown that at sufficiently large separations, U varies as r7 rather than rr6. In general, retardation can be allowed for by applying a correction function, f(#), to the London expression, whence where # = 2nr/LC and Ac is the characteristic wavelength of the interaction. Although no simple expression for f (j4) exists, Overbeek 43 has given the following analytical formulae : U = - B r 6 f @ ) , (18) 0 < b < 3, 3 < b < a, f(b) = 1.01-0.14fi; f(b) = 2.45b-1-2.04jh-2.Overbeek and others ' 9 lo* 3 5 9 44 have used these formulae to integrate (18) and obtain what amount to retardation correction functions for Hamaker constants. If this procedure is carried out for a liquid film on a solid, then the following, rather lengthy, expressions are obtained for Aiif : 1 < 3;1,/2n, V1 6n13 Afif = -~ (dsz[ 1.01 - 0.28#" + 0.0143(p)3 - 0.0193(~")4] - d z z [ l . O l -0.28fi12+0.0143(#2z)3 -0.0193(#')4]]; (20a) V l 6n13 Aiif = - ---(d"'[:1.470(+is2)- ' -0.S16(#,")-2] -202 n-ALKANE FILMS ON CC-ALUMINA where flz = 2nZ/A",' and f i 1 2 = 2n2/ALz.Hence one may obtain expressions, in terms of retarded forces, analogous to (15), (16) and (17). In the fully-retarded limit, II = (dy- d%",'/79l4. (21) It has been argued 45-5 that the microscopic approach is, in many cases, inadequate for the calculation of dispersion forces between condensed media, and that the Lifshitz, macroscopic approach is both more general and more accurate. For a liquid film on a semi-infinite solid, the Lifshitz theory gives where a = ( X 2 - 1 + E ~ / E ~ ) and are those of the bulk solid and bulk liquid respectively, and are functions of the imaginary frequency icy = 2nivkT/k.The quantity Xis an integration variable and the summation index v takes values 0, 1,2, 3, etc. The prime on the summation symbol in (22) means that the term for which v = 0 must be multiplied by 3. As with the simplified microscopic approach, it is assumed that ( 8 v f / i h ~ ~ , , ~ = d , and Aff and IJ follow on the further assumption that (8Sf/8n9,,,f = s2. Although expression (22) is complicated, it can be evaluated numerically providing suitable data are available. The present author is grateful to Dr. P. Richmond 52 who has carried out this operation for n-octane and n-decane films on a-alumina. The calculations were simplified by the fact that each of these materials has one major absorption frequency centred in the ultra-violet. Consequently, below this frequency, it is possible to represent the dielectric constant of each material by a simple expression involving the characteristic frequency v, = c/A0 and the limiting dielectric constant in the visible region, co : = ( X 2 - 1 + 1 / & I ) .The dielectric constants and &,-1 1 + (t/2nv,)2' &(it) = l + Furthermore, by restricting calculations to relatively thick films (I -c 10 nm), it is possible to neglect contributions from frequencies greater than the absorption frequency. Both v, and E~ were determined in the manner discussed by Gregory 37 from the variation of refractive index with frequency. Necessary data were obtained from the literat~re,~,. 31 and the resulting values are listed in table 2 below. TABLE 2.-DISPERSION DATA a-alumina n-octane n-decane EO 3.056 1.927 1.967 v,/s-l 4 .1 6 ~ 1015 3.470 x 1015 3 . 4 2 9 ~ 1015 The calculated dependence of II on I for n-decane is shown by the solid curve superimposed on the data in fig. 5. Calculations for n-octane give disjoining pressures only 1 to 2 % higher. Agreement between theory and experiment is excellent, as can be seen even more clearly from the log-log plot in fig. 6. Presented in this way, the best straight line (by linear regression giving 96 % correlation) through the experimental points deviates from the theoretical curve by less than 2 % of I over the entire experimental range. In addition, the line has a slope of -3.77, as compared with a mean value of about -3.9 for the theoretical curve in this region.T. D. BLAKE 203 These slopes may be compared, in turn, with limiting theoretical slopes for fully retarded and unretarded forces of - 4 and - 3 respectively.The single, filled-in point in each of fig. 5 and 6 was obtained 20 min after exposing the system to more than 20 mrad of ionising radiation from an l3II source. The radiation had no significant effect on film thickness, and it was therefore concluded that electrostatic forces arising from tribolectric effects were not contributory to measured disjoining pressures. I , I , \ , \ IC 2C) 3 0 5 0 7 0 100 200 300 FIG. 6.-Log-log plot of the dependence of disjoining pressure, n, on film thickness, I, for films of n-alkanes on a-alumina. Data and theoretical curves as for fig. 5 ; - -, best straight line (96 % correlation) through experimental points.Fig. 5 and 6 also show disjoining pressure curves calculated on the basis of the microscopic theory, using eqn (16) and equivalent expressions derived from (20a) and (20b). Again, calculations are simplified by the simple dielectric properties of a-alumina and the n-alkanes, which allow the necessary Hamaker constants to be obtained in terms of v, and go according to the formulae given by Gregory 37 : and where vzl = 2v~v~/(v”,”vvl,l>. Again, the values of v, and co listed in table 2 were used. These gave for n-decane, atsz = 10.40 x J, and for n-octane, dS’ = 10.15 x It is clear from fig. 5 and 6, that over the range of film thicknesses studied, neglect J and &I1 = 5.68 x J and atzz = 5 . 4 0 ~ J.204 11-ALKANE FILMS ON CI-ALUMINA of retardation leads to large errors.It is also clear that when retardation is allowed for, the predictions of the microscopic theory are only slightly different from those of the macroscopic theory. Indeed, on the basis of the present data, it is not possible to distinguish which is the more accurate. It has been argued,53 that the simple additive approach of the London-Hamaker theory is not valid for condensed media because it neglects any perturbation of pair-wise interactions by neighbouring molecules. However, the above results suggest either that this objection is unimportant for dispersion interactions in the high frequency region, or that any non-additivity is effectively accounted for by the use of macroscopic dielectric properties, v, and E ~ , to calculate the Hamaker constants.In either case, a significant part of the success of the microscopic approach with this system can be attributed to the simple dielectric properties of the materials involved. With more complex dielectrics, such as water, in which orientation and inductive effects are important, the microscopic approach appears to be much less success- As noted in the Introduction, measurements of disjoining pressure and vapour phase adsorption are complementary. It is therefore of interest to compare theory not only with the present results, but also with some earlier data 54 for much thinner films obtained by gravimetric measurement of the adsorption of n-decane on the oxidised surface of aluminium foil. For the purposes of such a comparison, ATf is a more appropriate parameter than II.Accordingly, fig. 7 shows a log-log plot of the theoretical dependence of -Aff on I for I between 0.1 and 100 nm. On this extended scale, it can be seen that the curves for the retarded region merge with the line for un-retarded forces at I = 5 nm. Superimposed on the theoretical curves in fig. 7 are the experimental points. The lower group were derived from the disjoining pressure measurements for decane using eqn (9), taking v z = 1.95 x m3 mol-l. The upper group were derived from the adsorption data using (10) and on the assumption that I = u'nf/Af. B.E.T. treatment of the adsorption data gave 0.72 nm2 for the molecular area of n-decane and 0.45 nm for the thickness of the first monolayer. The latter value has been used to express I in terms of an equivalent number of monolayers for the top scale of fig.7. However, the scale is somewhat artificial, since the molecular area is consistent with molecular orientation parallel to the surface,55 and this is unlikely to be preserved much beyond the first layer. For films of between 1 and 5 monolayers, the adsorption data lie reasonably close to the theoretical line and have a similar slope. In fact, theory underestimates I by about 15 %. Electron diffraction 56* 57 indicates that the oxide layer on aluminium is amorphous, and hence has a slightly lower density and dispersion interaction than a-alumina. At worst, this would lead to a further 10 % underestimate of I. Such discrepancies could be explained by a comparable underestimate of adsorbent area, as this was determined 5 8 by the B.E.T. method using an adsorbate (tetramethyl- silane) having large, approximately spherical molecules (area 0.46 nm2), and gave a roughness factor of only 1.08.This value is 16 % lower than that obtained by Bowers 59 using nitrogen (area 0.162 nm2). An alternative possibility is that the extra adsorption was due to the substrate aluminium. However, the alumina layer resulting from oxidation at normal temperatures is at least 5 nm 57 so any contribution from imaging forces should have been small. One additional consideration is that the alumina layer has a small surface dipole which may be expected to generate induced dipole interactions with at least the first monolayer of decane. Taking a value of 5.7 V nm-1 for the surface field,60 and 8 x nm3 for the polarisability of the n-decane molecule in the direction normal fu1.469 48, 51T.D. BLAKE 205 to the Clo chain,61 and ignoring lateral interactions, one arrives at a contribution to ASf of about - 8.8 x lo3 J mol-1 at monolayer coverage. As this is about one and a half times the calculated contribution from dispersion forces, one would expect the experimental dependence of Aff on I shown in fig. 7 to become steeper in this region. In fact, the opposite occurs, with Aff declining increasingly slowly as zero coverage is approached. One possible explanation is that the induced dipole contribution is being submerged in compensating for the obvious inadequacies of the present calcula- tions for films of molecular thickness.The identification of I in these calculations with v'nf/Af is one such inadequacy, and is probably the initial reason for the sharp divergence between experiment and theory below monolayer coverage. number of statistical monolayers f/nm FIG. 7.-Log-log plot of the dependence of differential free energy of film formation, Aff, on film thickness, 1 : 0, n-decane on a-alumina calculated from the experimental data of fig. 5 ; A, n-decane 011 oxidised aluminium foil calculated from vapour adsorption measurement~.~~ Curves indicate the theoretical contribution of dispersion forces to Aff calculated according to : -, Lifshitz theory ; - -, London-Hamaker theory ; - - - -, London-Hamaker theory, but allowing for retardation. Whilst more appropriate calculations for the monolayer region are beyond the scope of the present work, it is evident that they can only follow detailed knowledge of the material and electrical properties of both the film and the solid surface.In particular, the dependence of the molar entropy of the film on its thickness can no longer be neglected, and it must be decided whether one may treat a film of molecular dimensions as a continuum or whether one must, of necessity, consider individual molecular interactions. For filiiis of more than 5 monolayers, fig. 7 shows a gradual divergence between theory and experiment. However, over this regionp'/p; > 0.96, and it is therefore very likely that an increasing part of the measured adsorption was in the form of206 n-ALKANE FILMS ON Q-ALUMINA interlaminar condensate within the adsorbent sample.* In consequence, there seems no reason to doubt that dispersion force theory adequately accounts for the inter- mediate range of film thicknesses not investigated experimentally. CONCLUSIONS (i) n-Octane and n-decane form stable wetting films on a-alumina. (ii) The combined use of disjoining pressure and adsorption measurements provides a convenient way of investigating the properties of these films over a very wide range of film thicknesses. (iii) Both Lifshitz and London-Hamaker theories of dispersion forces correctly predict the disjoining pressure and differential free energy of formation of the films, provided the films exceed monolayer coverage and provided due allowance is made for retardation if the films exceed a thickness of about 5 nm.(iv) In making these predictions, it is sufficient to assume throughout that the films have molar volumes, molar entropies and dielectric properties identical with those of the parent bulk liquids. SYMBOLS area Hamaker constant London constant index for bubble speed of light (2.998 x lo8 m s-l) image density Helmholtz free energy index for film differential Helmholtz free energy of film formation Planck constant h/2n intensity of reflected light intensity of incident light general numerical indices (i # j ) Boltzmann constant film thickness index for liquid Avogadro’s number number of moles refractive index pressure saturation vapour pressure 2n3/AC in (1 8) and (19) ; 27d/Ac in (20) gas constant radius of curvature of bubble at its base radius of ith or jth interference fringe intermolecular distance J-1 4 S s S T U Ailf V 0 V X a! D 6 &O A.& A, P v, V i t IT P P e n 0 Fresnel coefficient in (1 1) entropy molar entropy index for solid temperature internal energy differential internal energy of film formation volume molar volume index for vapour integration variable in (22) ( x2 - 1 + &S/&I) ( x2 - 1 + 1 / E l ) 4niwf2/A. in (1 1) dielectric constant limiting dielectric constant wavelength characteristic dispersion wave- length chemical potential summation index in (22) characteristic dispersion frequency imaginary part of complex frequency disjoining pressure 3.1416 reflectivity of film correction term for extraneous light in eqn (12) interfacial tensionT. D. BLAKE 207 The author gratefully acknowledges many helpful discussions with Dr.J. F. Padday, and is especially grateful to Dr. P. Richmond for carrying out the calcula- tions involving Lifshitz theory and for making the results available for publication. B. V. Derjaguin and M . Kussakov, Acta Physicochim., 1939, 10, 25, 153. €3. V. Derjaguin, M. 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