J.C.S. Faraday I, 1980,76, 118-125Use of the Gibbs Equation to Calculate Adsorptioninto Monolayer-covered SurfacesBY DAVID M. ALEXANDER* AND GEOFFREY T. BARNESDepartment of Chemistry, University of Queensland, Brisbane, AustraliaReceived 5th December, 1977A formal equation has been derived to describe the change in equilibrium surface pressure whenthe concentration of a soluble surfactant is varied in a subphase covered by a constant amount ofinsoluble monolayer on a fixed area of surface. Approximations are suggested which enable theequation to be applied to experimental data for the calculation of the surface excess concentration ofsurfact ant.An example is given and the surface penetration results discussed.The adsorption of a solute at an interface can, in principle, be calculated fromsurface tension data by means of the Gibbs equation.However, in penetrationexperiments, where adsorption is occurring at a surface that carries an insolublemonolayer, there axe difficulties with such a calculation because the activity of themonolayer component in a bulk phase can be neither controlled nor measured.has attempted a solution to this problem, but his treatment containsassumptions which will be shown to limit the validity of the final equation to restrictedregions of high surfactant concentration and low monolayer coverage. In the presentpaper we have derived a more general expression which illuminates the assumptionsof the Pethica equation, defining the limits of its applicability, and which, usingcertain assumptions, may be applied to certain systems to give results for conditionswhere Pethica's equation is not applicable.PethicaTHEORETICALFor simplicity we will consider the case where both monolayer? and surfactantare uncharged.We will adopt the Gibbs convention €or defining the position of thesurface. For this situation the Gibbs equation can be written :lwhere rM is the adsorption of the insoluble monolayer substance and rgl) the adsorp-tion of the surfactant relative to water. AM is the absolute activity of the insolublemonolayer in the surface phase and As the absolute activity of the surfactant in boththe surface phase and the bulk phase at equilibrium.By definition, the axea per molecule of M isdn[ = RTTL d In AM + R T r p d In A:, (1)(2)(3)h AM = A/nh = 1 / r M .Furthermore, if the surfactant concentration is low we can write :d In As = d ln ms,? The term " monolayer " is used here to refer to the spread monolayer substance, M.11D.M. ALEXANDER AND G. T. BARNES I19where m, is the molality of surfactant. Inserting eqn (2) and (3) into eqn (1) gives(4)RTdIT = - d In AG+RTal) d In mS.A MIn an equilibrium penetration experiment, surface pressure isotherms are deter-mined on surfactant solutions of various concentrations. A plot of surface pre2sureagainst the logarithm of the surfactant concentration at a constant value of AM isthen constructed and the slope determined. Eqn (4) gives an expression for the slope :(E) = RT -(-) a h a , +RTr-I)a In mS A,ng AM a In mS A,ngRT a h & - - - ( - allI ) (s) +RTTh1),JM 2x mS A,ngwhich on rearrangement yieldsPARTIAL MOLAR AREAIt can be shown that Pethica's eqn 6(a), 9(u) and (11) are only self-consistent andconsistent with our eqn (5) ifwhere & is the partial molar area of M.However, the constant conditions ofeqn (6) do not correspond to those required for a rigorous definition of the partialmolar mea, as we will show.For a Gibbs model for the surfacedGz = -S" d T - A dy+x py dnt. (7)iAt constant T (a constraint not explicitly shown in the following expressions),andTherefore,The latter equality is in accord with the usual definition of 3,. Thus €or the penetra-tion experimentd In 1,AM = RT(T5r)ng,ng*It follows that eqn (6) is incorrect and cannot be substituted into eqn (5).Anotherapproach based on eqn (11) must be used120 ADSORPTION INTO MONOLAYERSRIGOROUS EXPRESSION FOR THE SLOPE, (anla In ms)A,"gAt constant T and n&, the activity of M will depend on the surface pressure andon the amount of adsorbed surfactant :For eqn (5) we require the differential with respect to II at constant area and n&.Thus, with the substitution of eqn (ll)y we obtainIf the last term in the denominator is zero the expression reduces to Pethica'sequation :SATURATION ADSORPTIONThe concept of saturation adsorption has proved useful in treating adsorptionat monolayer-free surfaces and it can also be applied to monolayer-covered surfaces.A suitable definition of saturation adsorption at a monolayer-covered surface wouldbe :Application of this condition to eqn (14) not only leads to the Pethica equation[eqn (15)], but also gives a constant value for the slope, (allla In ms)As ag.If, to thegeneral relation,A = n$&+n",A,,we apply the constant conditions of eqn (16) and the reasonable assumption thatAM and As do not vary with II in opposite directions, we see that AM, & and 2,must all be constant at saturation adsorption and that consequently= constant.Linear portions often observed 1* in plots of n against In m, are thus likelyto be a necessary condition for saturation adsorption, but because of experimentalinaccuracies not a sufficient condition. Within such a region the Pethica equationis valid and can be used to calculate the adsorption.For this calculation a value of AM is required.Pethica suggested that a reason-able estimate could be obtained by equating AM to the area per molecule in a mono-layer of pure M at the surface pressure selected for measuring the slope (alI/a In ms)A, nsD . M. ALEXANDER AND G . T . BARNES 121This assumption appears acceptable if the excess surface region is dilute with respectto surfactant. Fortunately the c?lculated adsorption values do not depend stronglyon the value used for AM unless AM is small.The appearance of linearity in the plot of Ii against lnm, may be deceptive,particularly at low areas where there is often appreciable scatter in the data. Insuch doubtful cases it is better to use the more general equations developed below.GENERAL CASEFor a more general treatment it is convenient to modify eqn (14). By definitionA = n&& = n;/r$'),&ril) = ngln;.whenceCombination with eqn (17) yieldsand eqn (14) can then'be written(20)Approximations are necessary if this equation is to be used to evaluate and &.(i) APPROXIMATION FOR (a In LM/an& ng.We assume that at constant surface pressure and temperature :This postulates ideal dilute solution behaviour for the excess surface components,the monolayer acting as solvent and the surfactant as solute, and can be assumedvalid only where ng 4 n;.It is assumed that under these c nditions the surfactantmolecules are molecularly dispersed in the excess phase. (The formation of twophases in the surface region would result in the loss of a degree of freedom whichwould be experimentally observable.)This equation applies when the surfactant is unionized.In the case where thesurfactant is ionized and surface hydrolysis does not occur the expression is morecomplex but tends to a value twice that given by eqn (22) as n:ln& + 0.{ii) APPROXIMATION FOR (6nz/aII)As ng.Differentiating eqn (17) with respect to II, keeping n& and A constant, giveswhenc122At constant nkADSORPTION INTO MONOLAYERSthereforeSimilarlyThe Gibbs-Duhem equation applied to the areas isSubstituting the above three equations into eqn (23) ;Provided that n: is appreciably less than n s an approximation for the last term ineqn (25) is acceptable.We shall assume that the partial compressibilities of thetwo components are equal, i.e.,Substitution into eqn (25) givesCALCULATION OF ;IsFor n$ 4 n&, which is a condition most frequently encountered when AM is small,we can combine eqn (22), (26) and (20) and using eqn (17) obtain :AM and (azM/aII) may be evaiuated by assuming, with Pethica,l that their valuesin the penetrated monolayer are the same as in a monolayer of pure M at the samesurface pressure. Eqn (27) can then be solved for As and eqn (19) gives the adsorp-tion, rp.The assumption of the constancy of & at constant I3 implies by the Gibbs-Duhemeqn (24) that As is also constant. Any treatment postulating a varying AM woulduse the Gibbs-Duhem equation, but experimental accuracy has not been high enoughto justify any such treatment.In the general case when AM is not lage it is apparent that a value of As can becalculated only if a very precise value of d?fM in the pure monolayer has been deterD.M. ALEXANDER AND G . T. BARNES 123mined, However, if the data were accurate enough AM and As could each beevaluated as constant parameters giving the best fit to the experimental data.In the case of a highly incompressible monolayer the situation is more favourablesince, because (i3&/i31X)ng, nE is small, the quotient in the brackets of eqn (27) is smallat all but very low surface concentrations of surfactant. Thus the approximationsmade become less critical if the monolayer is highly incompressible.There is another set af conditions which allow a solution for eqn (20). This iswhere eqn (20) reduces to eqn (19, Pethica’s equation, so that As can be calculatedusing eqn (19).The conditions are a Lasge value of AM [so that AM-& is large ineqn (20)] and bulk surfactant concentrations which are not too low [so that (i3nXldII), n%is small in eqn (2011.DISCUSSIONThe data of Pethica at 298 K on the penetration of cholesterol monolayers bydodecylsulphate can be treated with some confidence over an experimentally accessibleconcentration range because the cholesterol layer is so incompressible. We assumethat surface hydrolysis of the surfactant does not occur, though this has been verifiedfor the pure aqueous surfactant system only down to a concentration of 4 x moldnn3, well above the concentrations used in Pethica’s penetration experiments.I I I ‘ 0 tQ.36 0.38 0.40 0.42 0.44area, A M l n m z mohuje-1FIG.1.-Experimental slopes for the cholesterol + dodecylsulphate system from Pethica at28 mN m-’ (0) ; slopes calculated according to eqn (28) (solid curves) for As/run2 molecule-l of0.50 (upper curve), 0.55 (middle curve) and 0.60 (lower curve); experimental slope for themonolayex-free system (broken line).A surface pressure of 28 mN m-l was chosen as being common to the greagstnumber of Pethica’s isotherms (n, In m,) each of which applies to a particular AM.The slope of each isotherm ( X I / a In ms)A,ng was determined at this surface pressur124 ADSORPTION INTO MONOLAYERSand plotted against &.From Pethica's isotherm for the pure monolayer, ai at28 mN m-1 was 0.369 x 10-1 m2 molecule-1 and= - 1.62 x m4 molecule"2.The appropriate form of eqn (27) for the case of an electrolyte solute undergoing nosurface hydrolysis and penetrating a layer with which it has no common ion is I-'. (28) (E) = ? F [ l - 2RT;Z,(a;l,/aJJ),,,,,ms A,nS (AIM - A M ) A M ( A M A M + As)A further approximation is introduced in the factor of two in the last quotient. Thisis discussed above following eqn (22). This approximation is again valid at lowsurface concentrations of surfactant.Experimental and theoretical values are compared in fig. 1 which includes thepoint ( X I / a In ms)A, ng = 0 where A, = &. Plots are included for As = 0.50,0.55 and 0.60 nm2 molecule-'.The fit is best at the lowest adsorption (high mono-layer coverage), where the approximations are most valid, and for & = 0.55 nm2molecule-l.For the pure surfactant on water at 298 K the value of A, at surface pressuresabove 40 mN m-1 is in the range 0.41-0.58 nm2 molecule-l, indicating that thearea change on mixing monolayer and surfactant in the surface is not very large.Some estimate can be made of the error involved in this case when the assumptionfollowing eqn (25) breaks down. If, in the condensed S state the surfactant is assumedto be five times as compressible as the cholesterol instead of being of equal com-pressibility, then the maximum error resulting in the slopes calculated and plotted onfig.1 is 5 %, falling to 2 % at the lowest AM values used. A compressibility differenceof more than a factor of five would be unusual for pure monolayers in the S state.ACONCLUSIONIt has been shown that it is sometimes possible in practice to interpret penetrationisotherms by a thermodynamic treatment in terms of partial molar areas in regionswhere the surface is dilute in surfactant or in regions where the monolayer is of higharea per molecule and the surfactant is of relatively high concentration. In othercases other methods such as the use of radioactive tracers must be used to calculatethe surface excess of surfactant.The data of Pethica fit our equations as well as can be expected and lead to acalculated value of A, and the surface excess of surfactant at low surface areas ofmonolayers.The assumption is made that no surface hydrolysis takes place.Pethica's measurements of surface pressures over a subphase of 0.145 mol dm-3sodium chloride do not clarify this assumption, partly because of experimental un-certainties and partly because some measurements were made in more dilute solutionsmol d.~n-~) than used in the other penetration measurements, where the situationwith respect to surface hydrolysis may be different.Fowkes has discussed the problem of equilibrium penetration using a monolayermodel. His theory is based on a fundamental equation consistent with the treatmentabove but his assumptions regarding the ideality of the mixtures of the surface mono-layer are extended to high surface concentrations of surfactantD. M. ALEXANDER A N D G . T . BARNES 125NOTATIONarea of surface= A/$', area per surface excess mole of ipartial molar area of i defined by eqn (10)partial molar area of monolayer without surfactantGibbs energymolality of i in bulk phasesurface excess amount, Gibbs adsorption of component iAvogadro constantgas constantentropytemperature= np/Cnpsurface tensionsurface excess concentration (= nP/A)relative adsorption of component i with respect to component 3absolute activity of ichemical potential of isurface pressureisubscriptsM monolayer substances surfactantsuperscripts0 excess quantity referring to the Gibbs surfaceB. A. Pethica, Trans. Faraday SOC., 1955, 51, 1402.R. Aveyard and D. A. Haydon, An Introduction to the Principles of Surface Chemistry (Cam-bridge, London, 1973), pp. 12, 13.R. Defay, I. Prigogine, A. Bellemans and D. H. Everett, Surface Tension and Adsorption(Longmans, Green and CQ, London, 1966), p. 160.F. van Voorst Vader, Trans. Faraday SOC., 1960,56,1067.M. A. McGregor and G. T. Barnes, J. Colloid Interface Sci., 1976,54,439.J. E. Bujake and E. D. Goddard, Trans. Faraday SOC., 1965, 61, 19(F. M. Fowkes, J. Phys. Chem., 1961, 65, 355.(PAPER 712131