J . Chern. Soc., Faruduy Trans. 1, 1982, 78, 3319-3329 Thermodynamics of n-Alkane + Dimethylsiloxane Mixtures Part 4.-Surface Tensions BY BERYL E D M O N D S ~ A N D IAN A . MCLURE* Department of Chemistry, The University, Sheffield S3 7HF Receiued 17th February, 1982 Surface tensions have been measured by the differential capillary rise method under orthobaric conditions for binary mixtures of four n-alkanes (n-pentane, n-heptane, n-decane and n-tetradecane, replaced in the case of dimer by n-hexadecane) with four linear dimethylsiloxanes (dimer, trimer, tetramer and pentamer) at 303.2 K. The sign and magnitude of the excess surface tension depend ultimately upon the chain lengths of the components of the mixtures irrespective of whether a volume-fraction-based or a mole-fraction-based ideality is adopted.The dependence of the volume-fraction-based excess surface tension on chain length is very similar to that previously found for the excess volume. The results are analysed in terms of Prigogine's parallel-layer theory as modified by Gaines and Prigogine's average potential theory. Only the latter gives a useful description of the complexity of the chain-length dependence of the excess surface tension, and it is qualitative at best even then. We have reported previously measurements of gas-liquid critical temperatures and pressures, vapour pressures and excess enthalpies and volumes of mixing for mixtures of the type n-alkane+linear dimethy1siloxane.l Here we report the results of our measurements of surface tensions for sixteen of these systems and show that their complex dependence on the chain length of the components can be interpreted qualitatively in terms of a corresponding-states approach, not unlike that which was successful for the similar chain-length dependence of the excess volume.The sixteen binary mixtures studied under orthobaric conditions, i.e. at the saturation vapour pressure of the mixture at 303.2 K, comprised each of n-pentane, n-heptane, n-decane and either n-tetradecane or (for dimer only) n-hexadecane with each of hexamethyldisiloxane (dimer), octamethyltrisiloxane (trimer), decamethyl- te t rasiloxane (te tramer) and dodecame t hylpen tasil oxane (pen tamer). EXPERIMENTAL The orthobaric surface tension was measured by the differential capillary rise technique using a Pyrex glass cell containing only the degassed mixture.The diameters of the two Veridia precision-bore capillaries were confirmed at several points along the length of each capillary by weighing mercury. The diameters were found to be 1.00, and 0.19, mm (nominaIly 1.00 and 0.20mm, respectively) and the variation along the length of each capillary was 0.5%. The apparatus was cleaned successfully with permanganic acid, hydrogen peroxide and distilled water before oven drying. Mixtures were made up directly in the cell and their compositions were determined by weighing. The filled cell was wholly immersed in a water-filled thermostat whose temperature remained constant within & 5 mK. Equilibrium was deemed to have been reached when the difference in level of the menisci in the two capillaries, Ah, remained constant to within the precision of the cathetometer (kO.03 mm) for t Present address : Institution of Chemical Engineers, 3319 1 h.The surface tension y was Rugby CV21 3HQ.3320 n-A L K A N E + D I M E T H Y LSI LOX A NE MIXTURES where rl and r2 are the radii of the capillaries, g is the acceleration of free fall in our laboratory. and d is the density of the mixture calculated from the known densities of the n-alkanes2 and dimethyl~iloxanes~ and the excess volumes of mixing.' The contact angle was assumed to be zero; this was supported by visual observation, the good agreement of our values and literature values far the surface tension of the pure components and the similar agreement for the mixture benzene+n-hexane.* More details of the technique and of the materials used are available elsewhere.'.RESULTS Table 1 contains a comparison of our results for the surface tension y for the pure dimethylsiloxanes with some literature values; the agreement is within & 0.1 mN m-l in most caws. A fuller comparison will appear with the results of our measurements over a wide range of temperature.6 As a check on our procedure for mixtures we have measured y for benzene + n-hexane at 308.2 K. Fig. 1 shows our results and those of Schmidt et a1.;4 the agreement is satisfactory. The results of our measurements of y for n-alkane + dimethylsiloxane mixtures at 303.2 K are listed in table 2 and illustrated in fig. 2 as a function of the dimethylsiloxane mole fraction x,.The precision of y is believed to be 0.1 mN m-l, although as indicated above the accuracy may be less, and the precision of x, is - +0.001. TABLE 1 .-COMPARISON OF OUR SURFACE TENSIONS, y, FOR PURE SUBSTANCES AT VARIOUS TEMPERATURES substance T / K y/mN m-l y(literature)/rnN m-' Dimer 298 303 308 313 318 323 trimer tetramer pen tamer 298 303 308 313 318 323 298 303 308 313 318 323 328 333 298 303 313 318 323 15.4 15.0 14.7 14.3 13.9 13.5 16.6 16.2 15.8 15.4 15.0 14.6 17.3 16.9 16.5 16.1 15.7 15.4 15.0 14.7 17.7 17.4 16.7 16.4 16.0 15.7 (293 K)' 15.3 (297 K)', 14.82 (298 K)'O 16.96 (293 K)7 16.6 (297 K)9 16.05 (298 K)l0 17.60 (293 K)' 17.08 (298 K)1° 18.10 (293 K)' 17.7 (297 K)87g 17.08 (298 K)'OB. EDMONDS A N D I. A. MCLURE 332 1 FIG. 1.-Surface tension, 'J, for hexane (l)+benzene (2) mixtures at (from top to bottom) 303.2, 308.2 and 313.2 K: 0, this work; 0, according to ref (4).DISCUSSION Prigogine and Sarolea produced in 1950 the first molecular theory for the surface tension of chain-molecule mixtures,ll and successive refinements appeared over the next ten years. The most suitable data for testing the development of the theory have been the surface-tension results of Marechal for benzene-based monomer + dimer mixtures,12 those of Aveyard for binary mixtures of n-alkanes13 and those of LeGrand and Gaines for dimethylsiloxane oligomer + polymer mixturesg Each of these mixtures is of the kind which we shall call monohomologous, in that they contain only substances belonging to a single homologous series. The surface tension of such mixtures has an uncomplicated dependence on composition and chain length, and, not surprisingly, the behaviour can be fairly readily described by the Prigogine theory in either the original form or one of the modified forms.By contrast, just as with the bulk thermodynamics of chain-molecule liquid mixtures, the surface tension of polyhomologous mixtures, i.e. those containing substances from at least two homo- logous series, is more complicated than that of monohomologous mixtures and presents a greater challenge to theory. Thus our measurements of the surface tension of many of the binary dihomologous mixtures formed from the n-alkane, linear dimethylsiloxane and perfluoro-n-alkane series offer the opportunity for a more extended test of the theory of the surface tension of chain-molecule mixtures than has been available hitherto.Here we confine ourselves to the presentation and discussion of our results for n-alkane + dimethylsiloxane mixtures. From our previous work we know that the deviations from bulk ideality of these mixtures are modest, and we find that their surface behaviour is similarly only slightly non-ideal. The dependence on composition of the surface tension of all n-alkane + dimethyl- siloxane mixtures which we have studied is simple and regular. Only for n- pentane + dimer does aneotropy, or surface azeotropy, occur, and this probably owes3322 n-A L K ANE + D I M E THY LSI LOX ANE MIXTURES TABLE 2.-sURFACE TENSIONS, y, AT 303.2 K FOR n-ALKANE + DIMETHYLSILOXANE MIXTURES OF MOLE FRACTION X n-pentane n- hep tane n-decane n-hexadecane x, y/mN rn-l x, y/mNm-l x, y/mNm-l x, 7lmNm-l ~ 0 0.05 0.19 0.34 0.46 0.58 0.75 0.79 1 0 0.075 0.20 0.25 0.375 0.55 0.62 0.80 1 0 0.10 0.24 0.39 0.48 0.55 0.80 1 0 0.03 0.12 0.25 0.36 0.47 0.695 0.87 15.0 15.0 15.05 15.1 15.15 15.1 15.1 15.05 15.0 15.0 15.05 15.4 15.5 15.7 15.95 16.0 16.1 16.2 15.0 15.3 15.75 16.15 16.3 16.45 16.75 16.9 15.0 15.1 15.6 16.15 16.5 16.75 17.1 17.3 n-alkanes (1) + hexamethyldisiloxane (2) 0 19.3 0 22.9 0.08 18.7 0.07 22.1 0.19 18.3 0.16 21.1 0.37 17.3 0.295 19.75 0.55 16.5 0.42 18.7 0.67 16.1 0.61 17.4 0.80 15.65 0.75 16.5 0.88 15.45 0.90 15.6 1 15.0 1 15.0 n-alkanes (1) + octamethyltrisiloxane (2) 0 19.3 0 22.9 0.14 0.28 0.40 0.55 0.6 1 0.82 1 8.6 0.14 21.1 8.0 0.20 20.6 7.6 0.34 19.5 7.15 0.41 18.95 7.0 0.56 18.1 6.5 0.66 17.6 6.2 0.75 17.2 1 16.2 n-alkanes (1) + decamethlytetrasiloxane (2) 0 19.3 0 22.9 0.1 1 18.7 0.13 21.1 0.27 18.15 0.20 20.5 0.44 17.7 0.325 19.65 0.52 17.5 0.46 18.9 0.73 17.2 0.60 18.3 1 16.9 0.81 17.45 1 16.9 - - - - n-alkanes (1) + dodecamethylpentasiloxane (2) 0 19.3 0 22.9 0.05 19.0 0.04 22.2 0.13 18.7 0.12 21.3 0.27 18.3 0.32 19.8 0.40 18.0 0.49 19.0 0.515 17.8 0.67 18.3 0.58 17.7 0.76 18.0 0.77 17.5 1 17.4 0 0.175 0.28 0.38 0.55 0.60 0.76 0.86 1 0 0.20 0.25 0.44 0.55 0.66 0.70 1 0 0.18 0.255 0.33 0.44 0.61 0.71 0.745 1 0 0.14 0.18 0.34 0.52 0.65 0.78 1 26.6 23.2 21.6 20.5 18.7 18.3 17.0 16.2 15.0 25.7 22.1 21.5 19.65 18.75 18.0 17.8 16.2 25.7 22.3 21.5 20.75 19.9 18.8 18.3 18.1 16.9 25.7 22.5 21.9 20.4 19.4 18.8 18.3 17.4 more to the near equality of the surface tensions of the components than to any deeper cause; thus the occurrence here resembles a surface Bancroft point.The deviation of y from linearity in x is nowhere particularly marked; nonetheless we shall demonstrate that the dependence of this deviation from linearity on the chain length of the components is complex and provides the principal interest of the work. All of the foregoing contrasts sharply with the behaviour of mixtures containing perfluoro-B. EDMONDS A N D I. A. MCLURE 3323 0 0.2 0.L 0.6 0.8 1 .Y 2 288 26 I E Z E . - I c % . + 2 6 1 -1 0 0.2 01, 0.6 0.8 1 .Y 2 u I I r t l r l I I I 0.2 0.L 0.6 0.8 1 .Y 2 FIG. 2.-Surface tension, y , for n-alkane (l)+linear demethylsiloxane (2) mixtures at 303.2 K.In each diagram the curves correspond from bottom to top to n-pentane, n-heptane, n-decane and n-hexadecane (with dimer only) or n-tetradecane. Solid curves are drawn through the set of points corresponding to a given siloxane: 0, dimer; A, trimer; tetramer; '7, pentamer. n-alkanes, which is rich in non-linear surface-tension - composition relationships and in which both positive and negative aneotropy abound.14 The discussion of the surface thermodynamics of liquid mixtures is hampered by the lack of a statement of ideal behaviour enjoying the status of, say, Raoult's law or its near equivalents in the bulk thermodynamics of liquid mixtures. At least three3324 n-A LK AN E + D I MET H Y L S I LOX AN E MIX T U RES ways of describing ideality at the gas-liquid interface have been used.Two of these are essentially empirical and share with Raoult’s law the advantages of simplicity. They are ?id = x,Y,+x,Y, (2) and (3) where x i , 4i and yi are the mole fraction, volume fraction and surface tension in the pure state, respectively, of component i of the mixture and yid and yid are the surface tensions of the ideal mixture according to the two conventions. The second of these conventions is more appropriate for polymer mixtures. The third statement of ideality in currency was developed by Guggenheim from the quasi-crystalline treatment of interfaces;15 in this treatment the ideal surface tension of the mixture yg is given by exp(-yga/kT) =x,exp(-y,a/kT)+x,exp ( - y , a / k T ) (4) where a is the area of surface per molecule and T is temperature.This expression is obtained from the general theory by setting the interchange energy w equal to zero. This procedure is the counterpart of that which generates Raoult’s law from the quasi-crystalline theory for bulk mixture thermodynamics. The presence of the surface area a in eqn (4) introduces an undesirable lack of generality into this definition of surface ideality. The main attraction of the Guggenheim treatment is that it is based on a simple model and so it is easy to determine the effect of changes in the interchange energy on the surface tension of the mixture. The first two conventions are based on no model and it is hard to interpret in any simple way deviations from ideality so defined. Objections notwithstanding, each of these conventions possesses certain virtues of which we shall take advantage in turn.Our principle use of the concept of surface ideality is the definition of excess surface tension of surface ideality where the subscript indicates which of the three conventions for surface ideality is in Fig. 3 shows the excess surface tension for the equimolar mixture 7,” (x = 0.5) for all sixteen mixtures as a function of the n-alkane chain length n,. The most striking features are the regular family of smooth curves connecting quartets of points corresponding to mixtures containing a common dimethylsiloxane partner and the crossover point at n, = 6.5, which corresponds to the excess surface tension of an equimolar mixture of the hypothetical n-alkane of chain length 6.5 with any linear dimethylsiloxane. The magnitude of 7,” always increases with dimethylsiloxane chain length but only for mixtures containing n-pentane is 7,” positive.We see no reason to anticipate a cyclic pattern whereby positive y,” would recur at higher alkane chain length. In view of the relationship between mole fraction and volume fraction, it is no surprise to find regularities in the behaviour of the excess surface tension for the mixture of volume fraction 0.5 yf (4 = 0.5) shown on fig. 3 similar to that of 4E (x = 0.5) in fig. 4 although topologically different. Only for n-pentane dimer is yf positive, and the order of increasing magnitude of y,f on the high-chain-length side of the crossover point, in this case at nc z 7, is opposite to that for y f ; i.e.yf decreases with increasing dimethylsiloxane chain length for nc > 7. This feature and the generally smaller magnitude of yf compared with that of 7,” suggest that for chain- Play -B. EDMONDS A N D I. A. MCLURE 3325 1 ~ ~ ~ ' 1 ~ ~ 1 1 1 ' 0 - I E G 5 -1 - h m c I1 8 - - 2 -2 - 5 6 7 8 9 10 11 12 13 1L 15 16 'lc FIG. 3.-Volume-fraction-based excess surface tension, $, for n-alkane + linear dimethylsiloxane mixtures at 303.2 K and 4 = 0.5 plotted against the length, n,, of the alkane component of the mixture. Solid curves are drawn through the set of points corresponding to a given siloxane: 0, dimer; A, trimer; 0, tetramer; V, pentamer. ~ " " " " 1 ~ FIG. 4.-Moie-fraction-based excess surface tension, y:, for n-alkane + linear dimethylsiloxane mixtures at 303.2 K and x = 0.5 plotted against the chain length, n,, of the alkane component of the mixtures.Solid curves are drawn through the set of point corresponding to a given siloxane: 0, dimer; A, trimcr, 0, tetramer; V, pentamer. molecule mixtures yy is a more appropriate convention for surface ideality than ?id, at least in the predictive sense, and especially so when all components of the mixture are of large molecular weight. Both fig. 3 and 4 are very reminiscent of the diagram showing the fractional volume change at 4 = 0.5 for n-alkane+dimethylsiloxane mixtures as a function of nc [fig. 1 of ref. (lc)], in which the crossover points occurs at n, z 6.3326 n-A LK A N E + DIM ET HY LS I LOXANE MIXTURES Any successful theory of the surface tension of chain-molecule mixtures can reasonably be expected to describe and account for the following features of our experimental results and their counterparts in volume fraction terms.(1) The small positive 7,” of mixtures with n, < 6.5 and the increasing magnitude of 7,” of such mixtures with dimethylsiloxane size. (2) The existence of the crossover point in the neighbourhood of nc z 6.5. (3) The increasingly negative y,” with increasing dimethylsiloxane oligomer size at nc > 6.5. We describe next our attempts to achieve these aims in terms of two treatments of increasing sophistication. PR IGOG INE ’ s PAR A LLE L-L A Y ER MODEL TREATMENT A more realistic description of the surface tension of mixtures of molecules of different sizes was given by Prigogine and others, first for monomer+dimer and monomer + trimer, and then generally for monomer + oligomer of chain length r, called rmers by the Brussels school.A simplified model was later developed by Prigogine and Marichal for the special case in which rigid molecules only were considered and only those configurations in which the rmer lay parallel to the surface were taken into account.16 This treatment, based on the parallel-layer model, was developed chiefly for the athermal case, which we would expect to be a reasonable representation of n-alkane + dimethylsiloxane mixtures. The equations which yield the surface tension of the mixture are Y = Y1 + (W4 [In W 4 1 ) + (1 - r-l) (4; - 4211 where r is the chain length of component 2 of the mixture. A modification of these equations was suggested by Gains on the basis of a model in which only the surface layer behaves at her mall^.^ Non-ideality is allowed for by including a non-athermal interaction term in the chemical potential for the bulk mixture.The resulting generalised expressions for a mixture of oligomers of chain length rl and r2 are Y = Y 1 + ( W r 1 4 {In W 4 1 ) + (4; - 42) [I - Wr2)II - ( P / 4 4; = Y2 + ( W r 2 a) {In (4242) + (4; - 41) [(T2/Yl) - 11) -@/a) 4;. The following useful expressions for rl and r2 have been developed previously,lc where as before component 1 is the n-alkane and component 2 is the dimethylsiloxane: rl = (nc+2)/2 and Y, = 3nSi/2. The expressions are similar but not identical to those given by Simha and Havlik.” The surface area a was calculated from the volume per segment as a = (&/Nri)2/3 where Vi is the molar volume of component i and N is Avogadro’s number.The quantity p is in a sense an interaction energy similar to Guggenheim’s exchange energy w, but in Gaines’ treatment it assumes the role of a differential or excess interaction energy between the surface and the bulk of the mixture. In practice, p is an adjustable parameter which absorbs the consequences of effects neglected in the main theory. The computational procedure was to calculate 4’, the surface volume fraction, from the experimental y at 4 = 0.5, and then to calculate p at 4 = 0.5. Knowing B, 4’ was calcuated at each 4 and thus the entire theoretical isotherm was generated. The calculated and experimental isotherms are in good agreement.Typical resultsB. EDMONDS A N D I. A. MCLURE 2L * 3327 0 0.2 0.L 0.6 0.8 @2 FIG. 5. Predictions of the Prigogine-Gains parallel-layer model for surface tensions for the mixtures n-heptane +dimer (lower curve) and n-decane + pentamer (upper curve) at 303.2 K. The solid line represents the theory and the points are experimental. The dashed line corresponds to volume-fraction-based ideality. TABLE 3.-vALUES OF PARAMETER p REQUIRED TO DESCRIBE SURFACE TENSION OF n-ALKANE+ LINEAR DIMETHYLSILOXANE MIXTURES AT 303.2 K USING THE PRIGOGINE-GAINES PARALLEL- LAYER MODEL n-a1 kane dimer trimer tetramer pentamer n-pen tane - 5.8 + 9.4 6.4 5.8 n-heptane - 24.6 - 14.3 -9.3 -6.1 n-decane - 52.9 - 38 - 39.6 -38.1 n-tetradecane - 72.4a - 60.8 - 64.9 - 55.4 a n-Hexadecane + dimer.are shown in fig. 5 for n-heptane+dimer and n-decane+pentamer. Table 3 lists the values of p; they show some regularity with n-alkane chain length but they are independent of dimethylsiloxane chain length. This pattern can be interpreted loosely as reflecting the widely suspected difference between end- and middle-group interactions for n-alkanes, which is thought to be unimportant for dimethylsiloxanes and perfluoroalkanes. In Gaines' work on mixtures of toluene or tetrachloroethylene with dimethysiloxanes the values of p were considerably scattered, and little obvious correlation could be made with dimethylsiloxane chain length.'* The success of the model in describing our results is certainly to some extent a result of fortuitous compensation of errors.For example, deficiencies in evaluating the bulk-mixture chemical potentials are cancelled by errors in the surface-layer chemical potential. Only where the relative adsorption is weak, as in n-alkane + dimethylsiloxane mixtures, is the theory trustworthy, and even here its success is marred by the need to guess or otherwise put a value on p. Another objectionable feature of the method is that in no way does it account for the influence of chain flexibility on the surface tension of mixtures. In the next section we move to a discussion of a treatment which does attempt to introduce this effect.3328 n-A L K A N E -k D I METHY LSI L O X A N E MIXTURES PR I GOG I N E'S A VE RAG E-PO TE N T I A L THEORY In parallel with the extension to chain-molecule mixtures of the average-potential model (APM) for the bulk behaviour of mixtures of small molecules, Bellemans extended Prigogine's average-potential theory for the surface tension of mixtures of small molecules to mixtures of chain molecule^.^^ The essential features of the theory are that the surface tension of both components of the mixtures and of the mixture itself obey the same reduced surface equation of state, i.e.that the reduced surface tension yo2/&, where a and E are characteristic size and energy parameters, respectively, is a universal function of the reduced temperature_F= kT/E. The replacement of E , CT and by average values ( E ) , ( 0 ) and thus (7') generates the dynamic surface tension Ydyn of the mixture corresponding to a freshly formed surface of the same compcpition as the bulk of the mixture.If E and a for the components are not too dissimilar then ydyn and y of the mixture, sometimes termed Ystat, are related by In the form of the theory used here the usual APM rules for ( E ) and (a) were used. The reduced temperature was taken as ckT/qE* where c and q are the structural quantities related to entropy and energy in the usual corresponding-states treatment of chain molecules. The previous expressions were used for r, and r2 and the expressions of Simha and Havlik were used for c c, = ( n c + 7 ) / 6 and c2 = n,,+2. For q the following quasi-crystalline expression was used : Y = Ydyn - 4 2 ri r z a ? / 8 ( r ~ 41 -k r2 4 2 ) k T ] (&d~n/dd)~. qz = r ( z - 2 ) + 2 . At 303.2 K all our substances are far from their critical point, and so z = 12 was assumed.The surface area per segment of component 1, a;, was again estimated from a: = Vm(~egment)/~,N2/3. . 1 1 1 1 1 I I I ~ l 8 5 6 7 8 9 10 11 12 13 1G 15 16 "( FIG. &Predicted volume-fraction-based excess surface tension, y 8, for n-alkane + linear dimethylsiloxane mixtures at 303.2 K and 4 = 0.5 using Prigogine's average-potential model as extended to oligomer mixtures by Bellemans; nc is the chain length of the alkane component. The calculations incorporate the assumptions 6 = p = 8. From top to bottom along the left-hand side of the diagrams of the lines represent mixtures with second component in the order dimer, timer, tetramer and pentamer.B. E D M O N D S A N D I.A. MCLURE 3329 In the calculation the Progogine interaction parameters p, 6 and 8 were all set to zero. Patterson and Rastogi have demonstrated by phenomenological analysis that the n-alkanes and the dimethylsiloxanes obey the same surface principle of corresponding states20 and we have extended the certainty of this using new measurements of the surface tensions of the dimethylsil~xanes.~ The predicted surface tensions are in relatively pcor agreement with experiment, as can be seen from fig. 6 at least as far as magnitude is concerned. However, and we believe that this is more important, the variation of yf with chain length is qualitatively well-reproduced. Better agreement could undoubtedly be forced by adjustment of the various parameters in the equations, but in view of the well-known deficiencies of the average-potential model there seems little purpose in so doing.Note that the essential features of this model - the establishing of the various r, c and q average-parameter quantities - proceeded in the same way as for the similarly successful prediction of A V / V for the same mixtures. An alternative analysis of our results, free of the problems of the average-potential model, has been published by Dickinson.21 B.E. gratefully acknowledges receipt of a CASE studentship from the S.R.C. We both record our gratitude to Dr M. La1 of the Unilever Research Laboratory, Port Sunlight, for useful discussions during the course of this work. I E. Dickinson and I . A. McLure, J . Chmi. Sol,., Furudqj' Truns. I , 1974, 70. (a) 2313, ( h ) 2321 and (c) 2328. R. A. Orwoll and P. J. Flory, J . Am. Chem. Soc.. 1967. 55, 68 14. 1. A. McLure, A. J. 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Colloid Sci., 1952, 7, 122. l 7 R. Simha and A. J. Havlik, J . Am. Chtm Soc., 1964. 86, 197. IN G . L. Gaines. J . Phys. Chtwi.. 1969, 73. 3150. IY A. Bellemans, J . Chim. Phys., 1960, 57, 40. I" D. Patterson and A. K . Rastogi, J . PJ1y.s. Chcvn., 1970. 74, 1067. E. Dickinson. J . Colloid 1nrerfuc.e Sci.. 1975. 53. 467. (PAPER 2/301)