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Disjoining pressure in soap film thermodynamics
Coll~kis and Surfuces, 6 (1982) 241-264 Eketier Sdentific Publishing Cclmpany. Amsterdttm -
DISJOINING
PRESSURE
241 Printed in The Netherlands
IN SOAP FILM THERMODYNAMICS
JAN CHRlSTER ERIKSSON De~~tmentofPhyelcd (Sweden)
ChemisiFy. T/ie Rayal Institute of Technology_ 2004Q Stockholm
and BORISLAV V. TOSHEV Department
of Phydcal Chemidy.
Uttiueraity of Sofia, 1126 Sofia (Bedgarb)
(Received 2 April 1981:accepted in final form 7 June 1982)
ABSIRACIA rigorous and ewznttally complete thermodynamic ttiatment h presented of a thin soap film formed from an sdjaent men&us 6olutton. The entire map film BystemincIud@.~: the components: water (1).a sucfactant (2). a dt (3) and an inert gas (4). Mechanically. the thin soap fiImismodeled as two weakly inllteractlngaurfsces of tension positioned a distauce h c I- film thickness) apart. lir ia an auxiliary independent variable thet can be changed by’varying tlG+exte&l excess pressure acting in’ the normal direction an the surfaces of tension and counteracting the effective disjoining pressure, wf, u&bin the film.
The resuItaat excess in lateral tension within the fitrn fs assumed to be actItlg in the planes
at the surfaces of tenpIon. The thermodynamica of the fdm are basd on this me&anical model and are fomulnted la&h in-terms of the overall lilm tcnaion fr and the Rim surfa& tension 35; T&king due account of the vkiance of the soap llim system, a number of thermodynamic fundam&ntalequations are derived vrhicli ticrespond to different expaimental sitirhtionk Full con&ence with the thermodyuamiea of curve& fhaid interfacea ia catabllahed.‘It is noted that # and ‘cr aa wrellaa if8 and RD (the Derjaguin diajofning @rez~ sure) are ccnjugate t~rmodynamic quantities. ZrDaccounta for the tong range molecular interactionsbetween.thefilm faces in a clearcut fashion insofar as the aucfactant monolayer struk&~main essentialiy tiagfectcd by hr chang-za.@mtacf angie measurementi are particularly tifui &hen the c&f &ect is ti &cptare the th&modynamlc prO&ies of tbin soap films. For such measurements fit variationsare in genkl therm~dynamicaily inafgnifimnt for NB frbns but not far CB films_
242
is closely related to the earlier works of Toshev and Ivano~ 131 and Rusanov [4 ]_ In ail these treatments the disjoining pressure is a central quantity. However, there is still a certain fack of general agreement concerning the proper definition of the disjoining pressure in the cas? of a soap film and, to some extent also, concerning its more detailed fherzqc Dynamic significance, In a recent paper, Dejaguin and Churaev IS] tclrwarded a generally appliabte definition of the disjoining pressure as the ci’ffercnce between the normal pressure component, ph, in a thin ffuid fil:n at mechanical equilibrium rAd the (isotropic) pressure in the adjacent bulk t’:om which the film is formed. For equilibrium soap films, effectively the same definition has been used by I~usanov ES]. This kind of definition appears apprclpriate from a fundamental Iloint of view but still one might prefer an operatifJnal definition that is more directly derived from the experiment of separatinq the two film faces, i.e., on the disjoining experiment. One purpose of the present paper is to show that. the original disjoining pressure experiment with two interacting solid surfrIces may also serve as the conceptual basis for a disjoining pressure definition when treating a soap film, provided that two surfaces of tension are introduced which_represent the fiim mechanically. Along this line we here present a generalized but yet, as compared with the earlier treatments, simplified and rather straightforward thermodynamic analysis of mobile soap f’,Ims. Furthermore we scrutinize to what exteni terms of pV type actually need to be included in the resulting thermodynamic fundamental equations valid for a soap film at its equilibrium thi.ckness. FOP the sake of clarity, it is useful when dealing wit.h soap films to distinguish between the effective film disjoining pressure, rrf, that is closely related to the resultant film tension, yf. and the Derjaguin disjoining pressure, n psi_ At that in e corresponding way is associated with the film surface tension, +y a film thickness larger then the equilibrium thick+ ‘SS,-af equals the so-called driving force OP film thinning (that vanisiles at the equilibrium thickness) whereas zD is mainly d e termined by the (long range) interactions between the film surfaces and approaches zero when the film thickness becomes suffit-ientty large.
DESCRIPTION
OF THE
SOAP
FILM SYmEM
STUDIED
A horizontal, reiatively Ia~xo plane-parallel, mobk soap film which in the thermodynamic sense t compbtciy open to a surro~~hirig circular meniscus solution has been studied. For this choiceof film system geometry-we may neglect line tensiqn and gravity effecLs..fn the‘ mairi, the. soap film consists of a central water soIution core &th tine.moGkyex of aurfactant mtiieculees ad,vtbed a$ each of its fa&s. Thg phtiipa! radii of cur+& of the bicont-avo men&u5 are R i &nd R;: The men&& so!ution conkati tl~~$tiizompiindnts: .
213
water (component 1) an ionic Furfactant (component 2) and a salt (component 8). The ambient pressure is kept at p0 by means of a fourth, inert gas component, which is supposed to be insoluble in the meniscus solution. According to the ordinary phase rule, the entire equilibrium film system would have 4 degrees of freedom, In the present context, however, we also need to take into account variations of t-he mean curvature of the meniscus, implying that in total 5 degrees of freedom have to be specificed in order to completely determine the thermodynamic equilibrium state of the EiIm system [7]. Thus, one possible set of independent state varrables is T. pzr p3, p’, p,, where pI and p3 are the chemical potentials of the surfactant and the salt, respectively, and p* and p. denote the pressures inside and outside of the meniscus. Alternative sets of independent variables are T, p Ir pzr P 3rpa, and 2’. P,, pl, p3, p4, and T. x2, ICY,p’. p. where x2 and x3 arc the mole fractions in the meniscus sotution of the surfactant and the salt. For example, in a free equilbrium soap fiIm in contact with a meniscus of a fixed curvature kept at constant Par only 3 independent state variables remain: T, p2, px or T, x2, x3_ In particular, we may observe that for this kind of film system, the meniscus pressure p’ is n dependent variable as given by the Young-Laplace equation: p’ = P,, + ym( l/R,
+ l/R,)
= p. + ZymC
(1)
where rm(T, p2, p3) is the surface tension of the meniscus-gas interface and the me-n curvature of the meniscus_ R, and R1 we the (negative) radii of curvature of the biconcave meniscus.
C derides
THE STATSOF TENSION XNA SOAP FILM For a free, planar soap film (= for any planar interfaciat region) to be in mechanical equilibrium it is obviously necessary that the vertical (normal) pressure component, ph. is everywhere equal to the ambient pressure po_ The tangential local tension should largely vary with the z-coordinate perpendicular to the film pIane as is shown in Fig. I, It appears that in the monolayers, each of thq hydrocarbon chain regions supposedly yields a contribution to the resulting film tension of similar magnitude as the surface tension of a liquid hydrocarbon--air-interface. In the ionic head group regions, attractive forces, due to wat&-bydro=bon chain.contacts, are partially counterbaIanced by repulsive ioriic h$eractions (cf. 181). The diffuse double layer pai% should make rat&r small contributions to the overall fitm tensioll; yl’. t!lat can be related_+ the. usu@ way. by q?eans of the l&kker integral
to the tigetitial local tension, t (2) at distance z frosti the symmelzy plane of ; : __.-__ th&fti; ,;I ‘-.. .. _-
214
I~Z)404 d,"ckn?
Fig. L_ Schematic
local tension distribution
in the lateral direction far a thin
reap film.
Since the film is supposed to be planar and is surrounded by a gas phase at on both sides, we might formally attribute the film tension rr, when considered as the resultant excess in 104 free energy, to any geometrical surtiace parallel to the film and positioned within the film zone or its close vicinity. However, when yf is considered as the total excess in lateral tension, a largescale mechanical equivalence with the real film is realized when the surface of tension of the firm (which coincides with its symmetry plane) is chosen to represent the firm. But as the major contributions to yf arise in the monolayer regions at the film surfaces, a still closet mechanical correspondence with the real film may be attained by partitioning yf into two equal parts which are ascribed to a pair of surfaces of tansion located at z = H&/2 where ht is determined by the moment condition: p.
ht = (~/T!)J
0
(3)
-Ct(z1+I+&& .
-_
As is indiated
in Fig 1; we may assume that these two surfaces of tensionare positioned within the outer parts of the monilia~&ti-& the film surfaces. Hence; from a mechanical point of view, a soap’filiti’ is’analogous witha' pair of (interacting) plane-pa&let metibmnes of zero thickri&& each of lhich it; subject to a unifonri; isotropic tension ,Lf/2 that enclok a li@d at the Iaierai prekwe PO: According to’this -model d&&i&on,' the intihsiriical film thick-. ne&‘is defined as the disj&c&,h, bet&en thi- twb s&fai&s+f Wigion .&hich are aubsequentty treated +ti the di$dN &rfa&# b’ek&n~t%ti f iki tid the surrounding gas phase.
’
the surfactant and the salt are completely insensitive with respect to displace ments of the two dividing surfaces that fi~tmally enclose the film. In addition we suppose that the water vapour pressure is low enough to warrant that displacements within the film zone of the dividing surfaces between the film and the gas phase do not appreciably affect the value of r:. This is normally the case since the difference between r,’ as determiued by means of dividing surfaces corresponding to the film thickness ht and I?:, as determined using the mid-plane divi&nG surface, ia equal to hkc;’ whereas rf = &cl, c;’ and c; denoting the water concentrationsfmol/ml) in the gas phase and the meniscus phase, respectively. Hence, insofar as cy 4 c’, is fulfilled*, the vaIuc of r: is indeed insensitive to\=ds shifting the positions of the dividing surfaces within the film zone. This state of affairs implies that l?T can he measured experimeutally, e.g., through using fEf-laheiled water. Consequently, we introduce a thermodynamic film thickness variable, IQ,,, defined as hl,, = rF/c: which will be of use betow. This 1~~~)definition is illustrated in Fig. 2. c,n*rl &¶I
---h -+
----tA
: :-hl,,
:
2
Fig. 2. The thermodynamic film thtcknesslq,) = I.:~c; is defined so as to make -fit,2 c,(z)* = h(..)CL Le., the acels of different shades are of equal size. IhtB
Other supaficial Elm densities, in particular I?:. Sf/A and FE/A, vary slightly more w the positions of the diwiddingsurfaces are chmged. However, when choo&& the two surfaces of ten&ion at z = *lit/2 as tha dividirg surfaces it is reason&tile to &ii that,.approximately, I$ = 0 taking Pito account that the inert gas component, 4, is supposed neither to adsorb on.‘o the film surfaces n+r. to’.dis$olve in the aqueous core of the film. Thus, for simplicity, we ohdl_ hcriceforth utili.ze the_surfacm of .fenston at a.+ *ht/2 as the dividing surfaces t.hqt &jtib@ the fi.@. aqd. assume that. the condition r$ .= 0 is fulfilled for thischoice of pos@ms for-the dividing eurfacek AS shali tie verified Iater on, this s@p&fying assuojptgon will not greatly affect the result@~ fundamental _i equations:. --: .---L:.:-__ _’ :._ ._- c._ _:::
1’.
.
*At room tempelrtur&
c;ld~=3~10-s~
Furthermore, for the sake of clarity and extended validity, we take a generakEd appkach in the Eense that the film thickness is formally regzuded 83 an ad~it~~~al~ ~dependent variabb, To this end we may imagine that noninteracting plates with attached rods’are cocnected to the stretched membranes at z = sfzt/2 in the mechanical model representation (Pig. 3). By applying from tke outside, besides the press_~e pltYzxvariable force arA on the plates in the z-direction that counterbalances xhe force-5 tending to make the film thicker or thinner, the film thickness ht m&ht be ctiaqged in a quasi-static manner at fixed values of the environmental &&e variabkc, kg., at constant T, pl, pl, p’, pa_ nf is here called the effectpre film disjoining pressure. EvidentIy nf = 0 at the equi~b~um t~Iickne~ of a f:ee soap film.
Fig.3. By means of varying the effective film diGjoining preseura. q, the film thickness hc can he cfmmged in quajistattcmanner at coa+ant T, par g,, p’, pe. ‘CCcquat zero at the
equilibriumfitmth%&ness.
I
By proceeding in the manner described, equilibribrium soap fi!ms of a variable thickness at a fixed thermodynamic state as determined by, e.g., the independent St&e variables T, f12,p3, p’. pa will be covered by our treatment. The particular case of a film kept at its equilibrium thickness (of primary interest from the experimenta point OXview) is readily ubtained by putting at = 0 or by recognizing the fact that t‘ne thickness of such a film actually depends on the independent state variables, i.e., ht = ht (!I’, fl,, p ,, p!. PO). TfIE FREE
ENERGY
DIFFERENTIAL
OF’ THE SOAP
FILM
Referring to Fig. 3, we can now deducz the following expression for the Helmholtz free energy differential. dF’, v&d for an equihbs-bnr film of a variable thickness, dFf
= -SfdT-
Ipo +ar)Adht
+ (yf -poht)dA
+
& i=l
or,
pidn:
(4)
since Vf = htA,
dF’
=
-SfdT
1 pod%"
+
yfdA
3
- RfAdht -t C pidn: i31.
(5)
As already indicated above,-Ff, Sf, nf Ii = 1,2,3) are strictly defined by using the two surfaces of tension at z = ahJ2 as dividing surfaces for which the condition I$. = 0 is assumed to hold; Evidently, these film properties rather clo.uely correspond with the intrinsic thermodynamic film properties. Apart from the nrAdhr term added here, Eq. (5) is in full agreement with the free energy differential expression for a planar surface phase between two bulk phases as derived in detail by Guggenheim [lo]. Putting ITSequal to zero, Eq. (5) also agrees with the general free energy differential expression for a thin film at its equitibrium thickness assumed by Rusanov [G]. . By imroducing the-Gibbs free energy.of the film, here defied as, G’
= FE +&Jf
we get from Eq. (5):
(6)
dC’
3
= -mfAclilt + C
pidn’
(Z”,p,,A
constant)
_
(9)
i-1
From these expressions it is evident that whereas &A is the differential mechanic31 net work (other than ordinary pV work) associated with a roversible film arca increase, the corresponding mechanical net work in the direction normal to the film is -mfAdht. According to Eq. (7) we also have
=
-.ry
T,p,A,nf as a (mechanical) definition of the effective film disjoining pressure, Essentially the same basic drfmition might preferably be retained for all kinds of thin films irrespective of the nature of the proximate bulk phases (cf. [ $11). The corresponding (symmetric) film tension definition i3 given by the partial derivative
that constitutes the natural extension in this context of the commonly applied surface tension definition for ordinary, planar interfaces. We should note that Schetudko 1121 has employed the quantity uf defined by Eq. (10) in connection with kinetic investigations of soap films approaching their equilibrium thickness. In Schehxdko’s nomenclature -vf is caIIed the “driving force of thinning” though, as defined here, zf may equally well tend to make the film thicker (at sufficiently small ht). INTEGRATION
OF THE FREE
ENERGY
DIFFERENTIAL ‘.
Next we consider the process of form& additional soap film tirea by means of a stretching operation at constant T, prr &. p3; p,,, and ht. Mathematically, this process is accounted for by integrating Eq. (6) or Eq. (7), the result being &/A
=
FF/A +poht
3
= yf + c pir; I=1
:_ ’
An+her mode of integrafion_C b,&$ on a -w&i+ joining (tbin+fls) +qaer& men%,i.+ a thick film with’largq_&t G h
249
A comparison with Eq. (12) now shows that
This equation is fundamentally important since it states that the difference in the work of stretching/unit area between a thin and a thick soap film is equal to the reversible work of joining the two fiIm faces per unit area. The value of rf(@ depends on the exact value of hi and xf equalsp’ - p. = -Ap when ht is suffkientIy large. Thus rf(Jri) - rf(fzi) is normally (but not always)
THERMODYNAMIC
-drf
= (Sf/A)dT-hkdpo
+ ni dht + 6 I’fdMi jZl
(15)
that, as expected, is quite similar to the Gibbs surfnce tension equation. WE: note that all the si.~ differenti& dT, dpo,dhr, dpi (i = 1,2,3) are here considered as independer?& differentials and that the htdp, term may well yield a significant contribution to duf. By making use of the Gibkllluhern condition for the meniscus solution, Viz.,
S’dT+
a C
c;&i-
dp’ = 0
i=l
where A = S']V' and cf k n#' dpi
=-Z;dT
or, alternatively, the expression
+ ij;dp’ +
dxq
(i=
1,2,3)
(17)
,
Eq. (15) may be &written in terms of other sets of ikld&eitdent variables. In Eq. (17), g; -aYe 5 _e~ t& p=tg! *OF er+opy. and vc#+ne in tee meniscus solution; -On +aGnzi@rig $i; from Eq. ilS] by-u&g Eq_ (16) multiplied by hC,; Se obttriu
5
(c-i’- h
14.2 ;. i . --_
c;)c4q
-hi&&)
,:. ;
- 2&,;&, (18)
where the identity hCt)ci = l?f has been inserted, Ap = p. -p’ and 2S,,, G ht - JtE,). Thus, for a soap film of equilitxium thickness (Rf = 0) studied at constant ambient pressure, po, and constant capillary pressure, Ap, the simplified (but exact) fundamentai equation d&f
=
is applicable by means of which S f/Aand ri, r! can be evaluated from yf is also a known function. data provided that h cIl = her, (T,p,.p(,) It is obvious that thep, dependence of -yf as given Eq. (18) ia rather weak since 6<,, is expected to be of the order of -10 A. Eq. (18) inchtdes the Frumkin relation [13) for a soap film of equilibrium thickness on the form
according to which an increase in the ca iMary pressure amounting to, e.g., 10’ dyne/cm* results in an increase in r P equal to 10” dyne/cm for a 1000 A thick film. Hence, we may conclude that Eq. (19) should be sufficiently accurate ako when analyzing moderately precise yf data with standard errors, say >lO-* dyne/cm, recorded at approximately constant & and Ap. However, such a degree of precision will generally be insufficient when the object is to elucidate the special properties of thin soap fiIms (cf. Fig- 5). , By introducing the expressions given by Eq. (17) in Eq. (15) we readily obtain a fuwiamental equation with T, ht. x1, x3, p’, and p. as the independent variables, viz., _d#
=
d7t2
T,P',Jf,
(21)
uw
(23)
av is expected to be rmaller than -h,: where hv = Ctci,, rfS; and 2+=/r, &CS1) as the voiume effects at the film formation are likely to be very smell.
For a film of equilibrium thickness we have, according to Eq. (23), the Frumkin relation (aTfia(AP)T,x,,x,,p,= hvThe fundamental equations (18).arid (21) conta%~ alternative definitions on derivative form of the effective disjoining pressure, Rf, viz., (24)
arc
t-1
aht T.=,.x,,P'.P~
= -Rt
(25)
that might tie cornpar& with Eq; (10). Equations (24) and (25) relate to in principle wssible disjoining experiments with soap films carried out at a constant themtadynamic state. On the other hand, Eq. (10) relates to a hypothetical disjoining experiment during which the film is closed. We may also point out thiit Eq. (24) or Eq. (25), upoti integ&tion, yields Eq. (14) in a s’xaightforward manner. Ash et al. 1111 have deducti an equation anabgue to Eqs_ (24) and (2s) for the ease of tw0 interacting solid surfaces exposed to an adsarbing gas. THE j?lLM SURFACE
TRNSIGN.
+.
AND
THE
D&tJAGUIN
DISJOINIXG
PRESSURE
The ovf?ralI film ten&m, rf, and the effective film disjoining pressure, of, are baS&alC~ defined &&~idi&es’~ thb la&ml &d normal dwctio& &spectively;‘ti&&th& &b&;;it;pre&*~~ o; -Anoth& g&ii@li$y’Is ip&zntly_ ti make use.of the men&us pr$asur? p’ as the refere_nce Ffe+re, inside t$G fi@+ In congr$&iii$‘~~~h’ &he sect d@nffi&n ‘of th& surface titisioti of !he meniscus we ~)L*,‘+ti~s int&S&i+ $i* yftim_&irf+t&si&i~ 7 fsS b$’ m&n& .. 1 of; . th& ,. relation ; ._
.* .-:;‘,.; ::__:-: :. .; -- -. :. : .+,+-h+jp. :;:I. ,;;, :j::__-:: . I -__I ,__-> -I. ,-:__ (26) . ‘,>:L;f’r,‘, :.;,1; ;_-.i... ;..-. :-__ : . .-- -_ -. .: : 1. _-“.T:: where, aa &fore, Ap den&es the cappressure PO’, p’:In q&i-thermadynamic terms, y ls is given by the expression L -:, -.;;.;. -,: : y : ,. -_.: #..s
Fig.4. Mechanical model desqiption of a thin soap Iilm in t&m of the f&x surface tension +fS, the Uerj~gu&i dis_iolning pre~ure =D and the meniscru premurep’.
-_
yfs = from which it appears the refereme pressure defined by yf =Zrfs + that ,j+aszz&ated@th m&en$ ~ort&ion
(27) that p* is the reference pressure for z < ht/2 and p,, is for z > h&Z. More correctly, yfs should rather be + hfgAp making use of the paxticular surface ob tension y H, the psition of whkh_is determined. by the ._ :.
-. .
Hmrever, on the b&s
-
of Eqs. (3) and. (28) it can be &own ,
,-
: _.-;-. _t:
that .-
.. :- -’ (29)
263
Fig.5. tC and 2tfs as functions of ha far a CB soap film at constant 9X23%1, pd. and meniscus composition (c, = IO-’ molll). It is assumed chat 6~ = S A _
layer repulsion and van der Weals attraction resuk in long range forces operating in the z-directiort in addition to the forces already accounted for by the pressure p’. It appears that the changes of ‘yf and 2-y” generated by varying ht may be very small on an absoIute scale, typically 5 lo-’ dyne/cm. The limiting slopes of yf and 27 fs at large ht are Ap artd zero, respectively. Proceeding in an analogous fashion in the dirt&ion perpendicular to the firm, i.e., making use ofpl. instead-of pa as the reference pressurc inside the film, we n-y write (cf. F&. 4) (30)
264
film equilibrium COnditUteS a direct measure of the disjOining pressure RD at this equilibrium state. We stress, however, thai the definition of m& as given ;~y Eq. (31) refers to a soap film in physicochemical equilibrium with a meniscus solution in an arbitrary thermodynamic state for which the mechanical equilibrium is supposed to be realized by compensating from the outside the force of thinning (or thickening) acting in the z-direction, i.~., it applies to an equilibrium film of arbitrary thickness formed at given vales of T, pi, ~1~.p’, paEssentially the same “D definition has frequently been applied in the past when evaluating =D from kinetic measurements on the thi;ming process which enable the determination af mf. When ht becomes largeenough, ITS= -Ap,and hence mD = 0. Thus it is clear that mD depends on the intei:iction between the two surfaces of tension which mechanically represent the soap film. In Fig. 6 the aD
*f
= -S’dT
-p’dVf
+ 2rfsdA
- nDAdht
+ 6
y@n:
(33)
I=1
Introducing the Gibbs free energy of the film this time as Gf’ = Ff the corresponding Gibbs free energy differential becomes dGf’
+p'Vf,
3
= -SfdT+
Vf&s'+2~f8dA-aDAti~t
+ C
(341
p@nf
i-1 Thus, the partial derivatives
(%yTp_ * . , A
ilGf’
dA T,p*,h.nf (-_)
mz
=
=
(35)
-”
2gs
(36)
provide aIternative definitions of fly and yf*. The integrated relation resulting from Fq. (33) is Gf* F' -= A A
+
prht
which leads to the subsequent muation of the Gibbs-Duhem _zd#"
=
dT-
htdp’
+ nDdht
+
3 C
i-l
kind
$dp,
But in this equation only four of the diffetintials dT, dp’, dpt (i = 1,2,3) ?re actually independent in view of the Gibbs-Duhem randition; Eq_ (16), that holds for the meniscus solulion. Making use of.Eq_‘(16) to eliminate dp,. from Eq. (38) we thus obtain the fundamental equation . -2d#”
=
Sf - h&)dT A_
+ nDdljt + i :.-
:
..
(39) ,_’
:
tt. follo~~from Eq. (39j that yfs and nb are interrelated accqrding to the .,.#+zp;.; _;’ -<- .. =::;._. :_- ., ;. -. :1 :- ..-_:_: : .! T,_-_‘_._-;. . :_z_-,;. . -_ .:_, I. ~, -! I ‘T_c-_: i__; -‘. ._.:;- ->-‘., :,-,::I
,_. _>=i, ,_. _.. .,..-. I ‘__, ,*+- :<_.., i_, .y
., _.
‘.
(41) where, in fact,thecanstraintofaconstantp’presaureisnotto
beneglected. The fundamental equation based on yfs with x2 and x3 as the composition variables, is the following as is moot readily verified by directly inserting the definitions of yfs and XE In Eq. (23).
-zS,dp’
(421
Although in the normal case, the contribution from the last 2+dp’ term in this equation is expected to be quite small it may still be of significance_ This. circumstance is exemplified in Figs. 5 and 7_ Figure 7 refers to a free equilib-. rium soslp film (xl = 0) where fat is varied by changing p’ whereas in Fig. 6, ht is varied by changing sr at constant T, x*, x3, p*;p,,; From Eq. (42): (43)
.
Thus -mD is twice the rate of change of 7% with ht at cqnstant state-of the meniscus sahtion as is hkewise_expressed, by-Eq. (41)‘*, : Uponcross-differentiating Eq. (42) we obtain ;_. . _. .’ aSv
a=23
(fp’ 1 _-
T.h.xa,X, .-;”
=-
( 1
_,
.
I
,._;
..
;-;
::
,._:
.
2-
aAtl:-. T.-xI.X,,p* , ;:. . . . :
I ‘.;_!-=
.
-.. :
!.
‘.
. ’ ._-
(44) _.
287
xxi
400
Jo0
6w
100
wo _-c--
‘coo
,‘c-u,
h;CA
_----c
//
/
_fl
rc-
-c--
Fig_7. Diagmm shotin the dependence of yc’ an he for CB soap fiIms of equilibrium thickness (SD = Ap) at constant 2’. p. and meniscus compoaitian but variable p’. The dashed paAt of the curves relale to negative Ap values. The same baaIc data have been wed as in Fig. I!_
268
Fig-S. A mechanical malogue of the soap CiIm system. ht may he varied by changing rf while simukaneo~ly adjusting the meniscus pi&on so ss to keepp’ con&ant. AItematively, l& can he varied keeping -f = 0 and changing p’. =p is represented by the force of a _ spring coupled to the surfaces af ten&n. I D is practicalty idependent of p’. i.e., r~ is a function of he only at con&ant T and meniscus composition. FUNDAMENTAL
EQUATIONS
ON DIFFERENCE
FORM
The experimental determination of the contact angle, 0, at the platiau border which separates the film from the meniscus, involves a direct compariI son- between the fitm surface tension rf!- and the surface tension of the meniscus, r” _-Methods are nof available which enable highly accurate measurements even of small (G 19) contact angles [ l&18]. At mechanical .equilibrium we obviously have rfs = y%os~ or :.
AT
.
7 ‘$6 -- r” .
f .r%Qit;g)
.._
..-
-1. : : .-.
. _
_::
-‘,_
__ : ...(45)
(46)
subscript (t; ¬ing surface excess relative to the surface of tension at t = h&2. Eq. (46) clearly agrees with the general form of the Gibbs surface tension equation valid for an interface of variable curvature insofar as the condition et =0 is fulfilled, U ext we wish to compare the real soap film with the idealized soap film formed under identical conditions and having the same thickness as the real film, i.e., ht = ht , ie_ wqabo presuppose that 6(,) = 6,,,+ implying that and rI = E‘,,ld. Introducing the notation ht,) i h(,&
(471 (i=
23)
etc., where, e.g., &, equals I’[ of a thick (hi) real film with nD = 9 diminished by (hi - &)c;, we can derive the following fundamental equation making use of Eq. (39) as the starting point:
-Zd(Ar)
= A
dT+nodht+
5 Al$dpi is2
With ht replaced by h<,) and written in ti less satisfactory form this equation has been~utilized previously by De Feijter et al. (2,361. It is presumably accurate.eno.ugh for most applications_ Even better suited for actual applicationa on coutact angle data is the correspondmg equation with T, ht, xa, xKj as the independent variables (cf. Eq. (42)): ,.
260
mined nD - ht functions recorded with T, x2, and 1c3as parameters. The relation between integral quantities a&ociated with Eqs. (48) and (49) is (cf. Eq_ (37)):
from erhich the free energy difference between the real and the idealized film is obtained. This equation was applied above when making the calculations for Figs. 5-7. Then Af’f , AI?: 70 was assumed to hold for the CB films considered and, moreover, we estimated the free energy difference A(F’/A ) from the expression (CGS units) [la] :
1tP2
- -+ 48,4,/c’; exp[-3.28.111’ 12mht2
fi&]
where the conventional terms are inserted for the free energy changes due to the van der W&s attraction and the doubfz layer’ repulsion_ For freesoap films of equilibrium thicI:ness studied at constant pa we have RD = Ap and ht = (T, x2. x3, Ap). Thus, when performing Ay measure me& at a fixed capillary pressure, Ap, and constant ambient pressure pa, the equilibrium thickness ht depends only on T, x2, x3. Under such circumstances contributions to the derivatives of Ay with-respect to T, x2, 1~~ will genemlly be obtained from_ the rf&ht term in Eq_ (49)_ For Newton black (MB) films, however, with ti typical equilibrium thickness of iat = 35 A 1171 these contributions appear. to be negligible since for such films ht is practically independent of T, x2. x3, and’dp. For common black (CB) soap films with I+ ranging between 50 and 10130A; the siruation is quit& different, Le., the equilibrium lhickuess is often strongly st&te dependent for these films. The data collected in Table 1 [14,X6] suggat that, as a rule, we can introduce the approximation dht (TV xKlrx3. Ap) = 0 for NB films whereas the Apdht term may well be a siieable term for CB films_’ .
--
CB film
CBfiIm’
NB film
hc=700A
he-200A
ht=35A
261 REMOVALPI?
THE CONSTRAINT
r’, = 0
The thermodynamic fundrrmental equations deduced in the previous sections include the simplifying assumption that -the film content of the gaseous component, is negtigiblu when the two surfaces of tension at z = Cht/Z are taken as the dividing surfaces which separate the film from the gas phase. For realistic “inert” gases such as He, Ar, N1, O2 this condition may not be futfilled with sufficient accuracy in view of the dubility of the gas in the! aqueous core of the film. It is hence of interea to investigate the consequences of rerrioving the constraint I$ = 0, Our starting point is then Eq. (15) or Eq. (38) supplemented with an additional rfdpl term, viz.,
rf ,
In order to transform these equations into forms corresponding to Eqs. (181 and (39) we make use of the Gibbs--Duhem conditions for the gas phase and the meniscus phase: ~“dT+c:‘d~, &dT+
eczdp,
&c;dpi-dp’ I=1
-dp, = 0
= 0
64) WI
with the purpose of eliminating the differenti& do, and dp+ The resulting equations are algebraidally c’o’niplex_However, if, as a more realistic approximation,‘Me assume that rf 7 ht,,c4, .- .' c: denoting the concentration ef the gasc&S’&&&ent ‘dissolved in the m&is&s ph& and, furthkinoce, take into’zicc&irit H&t ,thoq&ofi&t (C~c;)/(c;c~) at r&m titmperatuti’ typidally i$-ieti th&‘IbLd for~~utibga& ti those Ii&d &ov& it is kadily’dem&retrated tb&tthe’f&ida&e&l equai&;na’ (l&&d ($9) h&v&d ptiviouafy (on &+n&g thad Tf 2 (@-rcm&-&f&i&t[y &u&e for vir$&@dl a&&&ions;’
262
-26J&’
(57)
By assuming that I$ = hC,& = 0 (the
and that the partial derivatives
meniscus solution is very dilute in component 4) the
double sums in the above equations can be reduced as follows: .&
*c2 {r~-&m,ci)(~k)
*k
T,P’,Xl
=
k$$
i$2(ri--
mc
T,p’,q
hk
Hence, a close correspondance with Eqs. (23) and (42) is achieved. The importit& fundamental equation5 on difference form derived in the previous section, viz., E+. (48) and (49), remain valid in unchanged form even when rf #Cl as is most readily verified on the basis of the exact Eqs. (53) and (57). Thus by assuming that I’: =l$ id and rf = rf iti when ht id equals ht. Eq. (48) is generated from Eq. (5b).‘If, in addition: we assum;? that &v=sv,td and that the cross derivatkfes (a&ax& 4 x x and are equal to zero, Eq. (49) is oi$k&d&om Eq. (57). @f131aX4)T,p-.xCt,~, SUMMARY
AND
CONCLUSiONS
The thermodynamics of thin acup filma as developed in the present paper is based upon the,conception that. two weakly interacting fluid intorkcial
283
solid surfaces submersed in a fluid phase. Also, the explicit introduction of thermodynamic excess properties of the soap fiIm reIative to the meniscus solution is deliberately avoided in order to gain simplicity. As a meniscus phase in necessarily part of the soap film syakm we may chose either the pressure, p’, in this phase or the gas phase pressure pa as the reference pressure inside the fiim when defining the film tension and the disjoining pressure. Thus the overall film tensio;l, 7’. and the effective film disjoining pressure, rf, represent the deviations from pn in the directions parallel with and perpendicuk to the seap film, respectively, whereas the film surface tension, yb, and the Derjaguin disjoining pressure, ng, accounts for the corresponding deviations relative top’. The main rer;ults LB!this investigation are embodied in the various fundamental equations deduced - Ew. (15). (1% W), WV, (4% 1421, $421, and (49) - which zarebased on either of the quantities TV, ~~~~or Ay ‘r s - 7m and are formulated in terms of both@ and x varisbles. All of the fundamental equations obtained include novel features iargeXy resulting from our fundamentaUy different approach as compared with the previous thermodynamic treatments of thin soap films available. In order to survey which independent state variables the characteristic thermoriynamic film quantities actually may depend upon, Table 2 below has been prepared where ++ indicates a strong or moderate dependence and + a weak dependence. We have indicated the consequences of the fact that A i$ for the surfactant compnent generally is expected to be a very small quantity IW. The Dejaguin disjoining pressure sD appears to be a convenient quantity from a theoretical point of view as in a rather clearcut way it reElecta the magTABLE
2
If
-.
-T
++
Ir,fx*
u,fxs
++
++
~~ (fef)*
++
++
rgr p (feE)
++
++
++
++
P’
++
PO
AP
ht
+
++
++
+
++
+
++
Y
++
++ +
I
264
nitude of the long range interactions between the two faces of the soap film at least inrofar as the monolayer structuresat the film surfaces remain essentially unaffected by & changer. But still, in the thermudynamics of soap films, the quantity 4 y undoubtedly plnys a more important role. It is also evident that it is only needed to introduce a disjoining pressurewhen dealing with thin films if the films are of such a nature that ht actualIy exhibits some appreciable state dependence or when ht can be directly varied by external agencies. With fairly modest changed the formal description of thin soap films given here should be appticable also for dispersed emulsion droplets. When discussing the stability of such systems it is evidently the effective disjoining pressure nf rather than mD that is of primazy importance. Concerning the experhnental possibilities at hand for exploring the thermodynamic properties of thin equilibrium soap film, it is clear that contact angle measurements arc particukrly useful. The alternative procedure to integrate mD - hl isotherms requires much more data collection for obtaining the same amount of thermodynamic information. When an extremely precise method to determine the film tension yf ia avaihble, e.g. by optically measuring the curvature of the film at a minute pressure difference across the film, direct yf measurements may also prove to be useful (cf. Eq. (19)). ACKNOWLEDGEMENT IVe wish to express our gratitude
to Dr. J.A.
ments on a preliminary versian of this paper.
de
Fejter for valuable com-
REFERENCES 1 2
3 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18
B.V. loerjaguinand M. Kussakov, lzv. AN. S.S.S.R., Ser. Khim., 6 (1936) 741. J.A. de Feijter, J.B. RIjnbout and A. Vrij. J. ColbId EnterfaceHci., 64 (1978) 268. B.V_ Toshev and LB. Ivanov, ColMd and Polymer %i., 263 (1976) 658,693. A.I. Rusanov, in B.V. Derjaguh (Rd.), Research in Surface Forces, Vol. 3, ConsuItanb Bureau, Plenum Press, New York, p. 111,197l. B.V. Derjaguin and N.V. Chutaev, J. Colloid Interface Sci., -66 (1978) 389. A-I. Rusanov, PhasengleichgetichL und Grr?nz~chenergcheinungen, Akademie-Verfg, Berlin, Chapter 14,1978. F-0. Koenig, J. Chem. Phys., 16 (1950) 449. J.N. Israelachvili, D-J. Mitchell and B.W. Ninham, Trans. Faraday SOC. II, 72 (197&) 1626. T.L. Hill, J. Phys_ Chem., 66 (1962) 526. E.A. Guggenheim. Thermodynamics. 6th Ed., North-Holland, Amsterdam, p. 46,1977. S-G. Ash, D.H. Eweett and .C..F. Radke, %bans. Faraday Sot. II, 69 (1973) 1266. A. 3chebrdko, Adv. CcUoid Ynterfaace Sci., 1,(1967) 391. A. mumkin, Zhur. Piz. Khim., 12 (1938) 337. J.A. de Feijfer, Contact Angles in Soap FiIms, Thesis, Utrecht, 1973. J.S. Cbnie, J.F. Goodman and B-T. Ingram, in E. Matijevic’(Ed.), Surface ana Corn!!! .. Science, Vot. 3,WUey Inter6cIence, BeW York, 1971, pi 167. J.A. de Feijter and A. Vrij. J. Calloid Interface Sci, 64 (1978) 269. ; n_ LA. de Feijter and A. Vrij, J. C$Md_Intcrfacs S&nce, 70 (1979) 466. R. Buscall and R.H. Oitenwill, in D.H. Everett (Ed.), CoIlaid ,V@, 2, &m. Chem. . .. Science, .. . _ . s43c., 1976.
DISJOINING
PRESSURE
241 Printed in The Netherlands
IN SOAP FILM THERMODYNAMICS
JAN CHRlSTER ERIKSSON De~~tmentofPhyelcd (Sweden)
ChemisiFy. T/ie Rayal Institute of Technology_ 2004Q Stockholm
and BORISLAV V. TOSHEV Department
of Phydcal Chemidy.
Uttiueraity of Sofia, 1126 Sofia (Bedgarb)
(Received 2 April 1981:accepted in final form 7 June 1982)
ABSIRACIA rigorous and ewznttally complete thermodynamic ttiatment h presented of a thin soap film formed from an sdjaent men&us 6olutton. The entire map film BystemincIud@.~: the components: water (1).a sucfactant (2). a dt (3) and an inert gas (4). Mechanically. the thin soap fiImismodeled as two weakly inllteractlngaurfsces of tension positioned a distauce h c I- film thickness) apart. lir ia an auxiliary independent variable thet can be changed by’varying tlG+exte&l excess pressure acting in’ the normal direction an the surfaces of tension and counteracting the effective disjoining pressure, wf, u&bin the film.
The resuItaat excess in lateral tension within the fitrn fs assumed to be actItlg in the planes
at the surfaces of tenpIon. The thermodynamica of the fdm are basd on this me&anical model and are fomulnted la&h in-terms of the overall lilm tcnaion fr and the Rim surfa& tension 35; T&king due account of the vkiance of the soap llim system, a number of thermodynamic fundam&ntalequations are derived vrhicli ticrespond to different expaimental sitirhtionk Full con&ence with the thermodyuamiea of curve& fhaid interfacea ia catabllahed.‘It is noted that # and ‘cr aa wrellaa if8 and RD (the Derjaguin diajofning @rez~ sure) are ccnjugate t~rmodynamic quantities. ZrDaccounta for the tong range molecular interactionsbetween.thefilm faces in a clearcut fashion insofar as the aucfactant monolayer struk&~main essentialiy tiagfectcd by hr chang-za.@mtacf angie measurementi are particularly tifui &hen the c&f &ect is ti &cptare the th&modynamlc prO&ies of tbin soap films. For such measurements fit variationsare in genkl therm~dynamicaily inafgnifimnt for NB frbns but not far CB films_
242
is closely related to the earlier works of Toshev and Ivano~ 131 and Rusanov [4 ]_ In ail these treatments the disjoining pressure is a central quantity. However, there is still a certain fack of general agreement concerning the proper definition of the disjoining pressure in the cas? of a soap film and, to some extent also, concerning its more detailed fherzqc Dynamic significance, In a recent paper, Dejaguin and Churaev IS] tclrwarded a generally appliabte definition of the disjoining pressure as the ci’ffercnce between the normal pressure component, ph, in a thin ffuid fil:n at mechanical equilibrium rAd the (isotropic) pressure in the adjacent bulk t’:om which the film is formed. For equilibrium soap films, effectively the same definition has been used by I~usanov ES]. This kind of definition appears apprclpriate from a fundamental Iloint of view but still one might prefer an operatifJnal definition that is more directly derived from the experiment of separatinq the two film faces, i.e., on the disjoining experiment. One purpose of the present paper is to show that. the original disjoining pressure experiment with two interacting solid surfrIces may also serve as the conceptual basis for a disjoining pressure definition when treating a soap film, provided that two surfaces of tension are introduced which_represent the fiim mechanically. Along this line we here present a generalized but yet, as compared with the earlier treatments, simplified and rather straightforward thermodynamic analysis of mobile soap f’,Ims. Furthermore we scrutinize to what exteni terms of pV type actually need to be included in the resulting thermodynamic fundamental equations valid for a soap film at its equilibrium thi.ckness. FOP the sake of clarity, it is useful when dealing wit.h soap films to distinguish between the effective film disjoining pressure, rrf, that is closely related to the resultant film tension, yf. and the Derjaguin disjoining pressure, n psi_ At that in e corresponding way is associated with the film surface tension, +y a film thickness larger then the equilibrium thick+ ‘SS,-af equals the so-called driving force OP film thinning (that vanisiles at the equilibrium thickness) whereas zD is mainly d e termined by the (long range) interactions between the film surfaces and approaches zero when the film thickness becomes suffit-ientty large.
DESCRIPTION
OF THE
SOAP
FILM SYmEM
STUDIED
A horizontal, reiatively Ia~xo plane-parallel, mobk soap film which in the thermodynamic sense t compbtciy open to a surro~~hirig circular meniscus solution has been studied. For this choiceof film system geometry-we may neglect line tensiqn and gravity effecLs..fn the‘ mairi, the. soap film consists of a central water soIution core &th tine.moGkyex of aurfactant mtiieculees ad,vtbed a$ each of its fa&s. Thg phtiipa! radii of cur+& of the bicont-avo men&u5 are R i &nd R;: The men&& so!ution conkati tl~~$tiizompiindnts: .
213
water (component 1) an ionic Furfactant (component 2) and a salt (component 8). The ambient pressure is kept at p0 by means of a fourth, inert gas component, which is supposed to be insoluble in the meniscus solution. According to the ordinary phase rule, the entire equilibrium film system would have 4 degrees of freedom, In the present context, however, we also need to take into account variations of t-he mean curvature of the meniscus, implying that in total 5 degrees of freedom have to be specificed in order to completely determine the thermodynamic equilibrium state of the EiIm system [7]. Thus, one possible set of independent state varrables is T. pzr p3, p’, p,, where pI and p3 are the chemical potentials of the surfactant and the salt, respectively, and p* and p. denote the pressures inside and outside of the meniscus. Alternative sets of independent variables are T, p Ir pzr P 3rpa, and 2’. P,, pl, p3, p4, and T. x2, ICY,p’. p. where x2 and x3 arc the mole fractions in the meniscus sotution of the surfactant and the salt. For example, in a free equilbrium soap fiIm in contact with a meniscus of a fixed curvature kept at constant Par only 3 independent state variables remain: T, p2, px or T, x2, x3_ In particular, we may observe that for this kind of film system, the meniscus pressure p’ is n dependent variable as given by the Young-Laplace equation: p’ = P,, + ym( l/R,
+ l/R,)
= p. + ZymC
(1)
where rm(T, p2, p3) is the surface tension of the meniscus-gas interface and the me-n curvature of the meniscus_ R, and R1 we the (negative) radii of curvature of the biconcave meniscus.
C derides
THE STATSOF TENSION XNA SOAP FILM For a free, planar soap film (= for any planar interfaciat region) to be in mechanical equilibrium it is obviously necessary that the vertical (normal) pressure component, ph. is everywhere equal to the ambient pressure po_ The tangential local tension should largely vary with the z-coordinate perpendicular to the film pIane as is shown in Fig. I, It appears that in the monolayers, each of thq hydrocarbon chain regions supposedly yields a contribution to the resulting film tension of similar magnitude as the surface tension of a liquid hydrocarbon--air-interface. In the ionic head group regions, attractive forces, due to wat&-bydro=bon chain.contacts, are partially counterbaIanced by repulsive ioriic h$eractions (cf. 181). The diffuse double layer pai% should make rat&r small contributions to the overall fitm tensioll; yl’. t!lat can be related_+ the. usu@ way. by q?eans of the l&kker integral
to the tigetitial local tension, t (2) at distance z frosti the symmelzy plane of ; : __.-__ th&fti; ,;I ‘-.. .. _-
214
I~Z)404 d,"ckn?
Fig. L_ Schematic
local tension distribution
in the lateral direction far a thin
reap film.
Since the film is supposed to be planar and is surrounded by a gas phase at on both sides, we might formally attribute the film tension rr, when considered as the resultant excess in 104 free energy, to any geometrical surtiace parallel to the film and positioned within the film zone or its close vicinity. However, when yf is considered as the total excess in lateral tension, a largescale mechanical equivalence with the real film is realized when the surface of tension of the firm (which coincides with its symmetry plane) is chosen to represent the firm. But as the major contributions to yf arise in the monolayer regions at the film surfaces, a still closet mechanical correspondence with the real film may be attained by partitioning yf into two equal parts which are ascribed to a pair of surfaces of tansion located at z = H&/2 where ht is determined by the moment condition: p.
ht = (~/T!)J
0
(3)
-Ct(z1+I+&& .
-_
As is indiated
in Fig 1; we may assume that these two surfaces of tensionare positioned within the outer parts of the monilia~&ti-& the film surfaces. Hence; from a mechanical point of view, a soap’filiti’ is’analogous witha' pair of (interacting) plane-pa&let metibmnes of zero thickri&& each of lhich it; subject to a unifonri; isotropic tension ,Lf/2 that enclok a li@d at the Iaierai prekwe PO: According to’this -model d&&i&on,' the intihsiriical film thick-. ne&‘is defined as the disj&c&,h, bet&en thi- twb s&fai&s+f Wigion .&hich are aubsequentty treated +ti the di$dN &rfa&# b’ek&n~t%ti f iki tid the surrounding gas phase.
’
the surfactant and the salt are completely insensitive with respect to displace ments of the two dividing surfaces that fi~tmally enclose the film. In addition we suppose that the water vapour pressure is low enough to warrant that displacements within the film zone of the dividing surfaces between the film and the gas phase do not appreciably affect the value of r:. This is normally the case since the difference between r,’ as determiued by means of dividing surfaces corresponding to the film thickness ht and I?:, as determined using the mid-plane divi&nG surface, ia equal to hkc;’ whereas rf = &cl, c;’ and c; denoting the water concentrationsfmol/ml) in the gas phase and the meniscus phase, respectively. Hence, insofar as cy 4 c’, is fulfilled*, the vaIuc of r: is indeed insensitive to\=ds shifting the positions of the dividing surfaces within the film zone. This state of affairs implies that l?T can he measured experimeutally, e.g., through using fEf-laheiled water. Consequently, we introduce a thermodynamic film thickness variable, IQ,,, defined as hl,, = rF/c: which will be of use betow. This 1~~~)definition is illustrated in Fig. 2. c,n*rl &¶I
---h -+
----tA
: :-hl,,
:
2
Fig. 2. The thermodynamic film thtcknesslq,) = I.:~c; is defined so as to make -fit,2 c,(z)* = h(..)CL Le., the acels of different shades are of equal size. IhtB
Other supaficial Elm densities, in particular I?:. Sf/A and FE/A, vary slightly more w the positions of the diwiddingsurfaces are chmged. However, when choo&& the two surfaces of ten&ion at z = *lit/2 as tha dividirg surfaces it is reason&tile to &ii that,.approximately, I$ = 0 taking Pito account that the inert gas component, 4, is supposed neither to adsorb on.‘o the film surfaces n+r. to’.dis$olve in the aqueous core of the film. Thus, for simplicity, we ohdl_ hcriceforth utili.ze the_surfacm of .fenston at a.+ *ht/2 as the dividing surfaces t.hqt &jtib@ the fi.@. aqd. assume that. the condition r$ .= 0 is fulfilled for thischoice of pos@ms for-the dividing eurfacek AS shali tie verified Iater on, this s@p&fying assuojptgon will not greatly affect the result@~ fundamental _i equations:. --: .---L:.:-__ _’ :._ ._- c._ _:::
1’.
.
*At room tempelrtur&
c;ld~=3~10-s~
Furthermore, for the sake of clarity and extended validity, we take a generakEd appkach in the Eense that the film thickness is formally regzuded 83 an ad~it~~~al~ ~dependent variabb, To this end we may imagine that noninteracting plates with attached rods’are cocnected to the stretched membranes at z = sfzt/2 in the mechanical model representation (Pig. 3). By applying from tke outside, besides the press_~e pltYzxvariable force arA on the plates in the z-direction that counterbalances xhe force-5 tending to make the film thicker or thinner, the film thickness ht m&ht be ctiaqged in a quasi-static manner at fixed values of the environmental &&e variabkc, kg., at constant T, pl, pl, p’, pa_ nf is here called the effectpre film disjoining pressure. EvidentIy nf = 0 at the equi~b~um t~Iickne~ of a f:ee soap film.
Fig.3. By means of varying the effective film diGjoining preseura. q, the film thickness hc can he cfmmged in quajistattcmanner at coa+ant T, par g,, p’, pe. ‘CCcquat zero at the
equilibriumfitmth%&ness.
I
By proceeding in the manner described, equilibribrium soap fi!ms of a variable thickness at a fixed thermodynamic state as determined by, e.g., the independent St&e variables T, f12,p3, p’. pa will be covered by our treatment. The particular case of a film kept at its equilibrium thickness (of primary interest from the experimenta point OXview) is readily ubtained by putting at = 0 or by recognizing the fact that t‘ne thickness of such a film actually depends on the independent state variables, i.e., ht = ht (!I’, fl,, p ,, p!. PO). TfIE FREE
ENERGY
DIFFERENTIAL
OF’ THE SOAP
FILM
Referring to Fig. 3, we can now deducz the following expression for the Helmholtz free energy differential. dF’, v&d for an equihbs-bnr film of a variable thickness, dFf
= -SfdT-
Ipo +ar)Adht
+ (yf -poht)dA
+
& i=l
or,
pidn:
(4)
since Vf = htA,
dF’
=
-SfdT
1 pod%"
+
yfdA
3
- RfAdht -t C pidn: i31.
(5)
As already indicated above,-Ff, Sf, nf Ii = 1,2,3) are strictly defined by using the two surfaces of tension at z = ahJ2 as dividing surfaces for which the condition I$. = 0 is assumed to hold; Evidently, these film properties rather clo.uely correspond with the intrinsic thermodynamic film properties. Apart from the nrAdhr term added here, Eq. (5) is in full agreement with the free energy differential expression for a planar surface phase between two bulk phases as derived in detail by Guggenheim [lo]. Putting ITSequal to zero, Eq. (5) also agrees with the general free energy differential expression for a thin film at its equitibrium thickness assumed by Rusanov [G]. . By imroducing the-Gibbs free energy.of the film, here defied as, G’
= FE +&Jf
we get from Eq. (5):
(6)
dC’
3
= -mfAclilt + C
pidn’
(Z”,p,,A
constant)
_
(9)
i-1
From these expressions it is evident that whereas &A is the differential mechanic31 net work (other than ordinary pV work) associated with a roversible film arca increase, the corresponding mechanical net work in the direction normal to the film is -mfAdht. According to Eq. (7) we also have
=
-.ry
T,p,A,nf as a (mechanical) definition of the effective film disjoining pressure, Essentially the same basic drfmition might preferably be retained for all kinds of thin films irrespective of the nature of the proximate bulk phases (cf. [ $11). The corresponding (symmetric) film tension definition i3 given by the partial derivative
that constitutes the natural extension in this context of the commonly applied surface tension definition for ordinary, planar interfaces. We should note that Schetudko 1121 has employed the quantity uf defined by Eq. (10) in connection with kinetic investigations of soap films approaching their equilibrium thickness. In Schehxdko’s nomenclature -vf is caIIed the “driving force of thinning” though, as defined here, zf may equally well tend to make the film thicker (at sufficiently small ht). INTEGRATION
OF THE FREE
ENERGY
DIFFERENTIAL ‘.
Next we consider the process of form& additional soap film tirea by means of a stretching operation at constant T, prr &. p3; p,,, and ht. Mathematically, this process is accounted for by integrating Eq. (6) or Eq. (7), the result being &/A
=
FF/A +poht
3
= yf + c pir; I=1
:_ ’
An+her mode of integrafion_C b,&$ on a -w&i+ joining (tbin+fls) +qaer& men%,i.+ a thick film with’largq_&t G h
249
A comparison with Eq. (12) now shows that
This equation is fundamentally important since it states that the difference in the work of stretching/unit area between a thin and a thick soap film is equal to the reversible work of joining the two fiIm faces per unit area. The value of rf(@ depends on the exact value of hi and xf equalsp’ - p. = -Ap when ht is suffkientIy large. Thus rf(Jri) - rf(fzi) is normally (but not always)
THERMODYNAMIC
-drf
= (Sf/A)dT-hkdpo
+ ni dht + 6 I’fdMi jZl
(15)
that, as expected, is quite similar to the Gibbs surfnce tension equation. WE: note that all the si.~ differenti& dT, dpo,dhr, dpi (i = 1,2,3) are here considered as independer?& differentials and that the htdp, term may well yield a significant contribution to duf. By making use of the Gibkllluhern condition for the meniscus solution, Viz.,
S’dT+
a C
c;&i-
dp’ = 0
i=l
where A = S']V' and cf k n#' dpi
=-Z;dT
or, alternatively, the expression
+ ij;dp’ +
dxq
(i=
1,2,3)
(17)
,
Eq. (15) may be &written in terms of other sets of ikld&eitdent variables. In Eq. (17), g; -aYe 5 _e~ t& p=tg! *OF er+opy. and vc#+ne in tee meniscus solution; -On +aGnzi@rig $i; from Eq. ilS] by-u&g Eq_ (16) multiplied by hC,; Se obttriu
5
(c-i’- h
14.2 ;. i . --_
c;)c4q
-hi&&)
,:. ;
- 2&,;&, (18)
where the identity hCt)ci = l?f has been inserted, Ap = p. -p’ and 2S,,, G ht - JtE,). Thus, for a soap film of equilitxium thickness (Rf = 0) studied at constant ambient pressure, po, and constant capillary pressure, Ap, the simplified (but exact) fundamentai equation d&f
=
is applicable by means of which S f/Aand ri, r! can be evaluated from yf is also a known function. data provided that h cIl = her, (T,p,.p(,) It is obvious that thep, dependence of -yf as given Eq. (18) ia rather weak since 6<,, is expected to be of the order of -10 A. Eq. (18) inchtdes the Frumkin relation [13) for a soap film of equilibrium thickness on the form
according to which an increase in the ca iMary pressure amounting to, e.g., 10’ dyne/cm* results in an increase in r P equal to 10” dyne/cm for a 1000 A thick film. Hence, we may conclude that Eq. (19) should be sufficiently accurate ako when analyzing moderately precise yf data with standard errors, say >lO-* dyne/cm, recorded at approximately constant & and Ap. However, such a degree of precision will generally be insufficient when the object is to elucidate the special properties of thin soap fiIms (cf. Fig- 5). , By introducing the expressions given by Eq. (17) in Eq. (15) we readily obtain a fuwiamental equation with T, ht. x1, x3, p’, and p. as the independent variables, viz., _d#
=
d7t2
T,P',Jf,
(21)
uw
(23)
av is expected to be rmaller than -h,: where hv = Ctci,, rfS; and 2+=/r, &CS1) as the voiume effects at the film formation are likely to be very smell.
For a film of equilibrium thickness we have, according to Eq. (23), the Frumkin relation (aTfia(AP)T,x,,x,,p,= hvThe fundamental equations (18).arid (21) conta%~ alternative definitions on derivative form of the effective disjoining pressure, Rf, viz., (24)
arc
t-1
aht T.=,.x,,P'.P~
= -Rt
(25)
that might tie cornpar& with Eq; (10). Equations (24) and (25) relate to in principle wssible disjoining experiments with soap films carried out at a constant themtadynamic state. On the other hand, Eq. (10) relates to a hypothetical disjoining experiment during which the film is closed. We may also point out thiit Eq. (24) or Eq. (25), upoti integ&tion, yields Eq. (14) in a s’xaightforward manner. Ash et al. 1111 have deducti an equation anabgue to Eqs_ (24) and (2s) for the ease of tw0 interacting solid surfaces exposed to an adsarbing gas. THE j?lLM SURFACE
TRNSIGN.
+.
AND
THE
D&tJAGUIN
DISJOINIXG
PRESSURE
The ovf?ralI film ten&m, rf, and the effective film disjoining pressure, of, are baS&alC~ defined &&~idi&es’~ thb la&ml &d normal dwctio& &spectively;‘ti&&th& &b&;;it;pre&*~~ o; -Anoth& g&ii@li$y’Is ip&zntly_ ti make use.of the men&us pr$asur? p’ as the refere_nce Ffe+re, inside t$G fi@+ In congr$&iii$‘~~~h’ &he sect d@nffi&n ‘of th& surface titisioti of !he meniscus we ~)L*,‘+ti~s int&S&i+ $i* yftim_&irf+t&si&i~ 7 fsS b$’ m&n& .. 1 of; . th& ,. relation ; ._
.* .-:;‘,.; ::__:-: :. .; -- -. :. : .+,+-h+jp. :;:I. ,;;, :j::__-:: . I -__I ,__-> -I. ,-:__ (26) . ‘,>:L;f’r,‘, :.;,1; ;_-.i... ;..-. :-__ : . .-- -_ -. .: : 1. _-“.T:: where, aa &fore, Ap den&es the cappressure PO’, p’:In q&i-thermadynamic terms, y ls is given by the expression L -:, -.;;.;. -,: : y : ,. -_.: #..s
Fig.4. Mechanical model desqiption of a thin soap Iilm in t&m of the f&x surface tension +fS, the Uerj~gu&i dis_iolning pre~ure =D and the meniscru premurep’.
-_
yfs = from which it appears the refereme pressure defined by yf =Zrfs + that ,j+aszz&ated@th m&en$ ~ort&ion
(27) that p* is the reference pressure for z < ht/2 and p,, is for z > h&Z. More correctly, yfs should rather be + hfgAp making use of the paxticular surface ob tension y H, the psition of whkh_is determined. by the ._ :.
-. .
Hmrever, on the b&s
-
of Eqs. (3) and. (28) it can be &own ,
,-
: _.-;-. _t:
that .-
.. :- -’ (29)
263
Fig.5. tC and 2tfs as functions of ha far a CB soap film at constant 9X23%1, pd. and meniscus composition (c, = IO-’ molll). It is assumed chat 6~ = S A _
layer repulsion and van der Weals attraction resuk in long range forces operating in the z-directiort in addition to the forces already accounted for by the pressure p’. It appears that the changes of ‘yf and 2-y” generated by varying ht may be very small on an absoIute scale, typically 5 lo-’ dyne/cm. The limiting slopes of yf and 27 fs at large ht are Ap artd zero, respectively. Proceeding in an analogous fashion in the dirt&ion perpendicular to the firm, i.e., making use ofpl. instead-of pa as the reference pressurc inside the film, we n-y write (cf. F&. 4) (30)
264
film equilibrium COnditUteS a direct measure of the disjOining pressure RD at this equilibrium state. We stress, however, thai the definition of m& as given ;~y Eq. (31) refers to a soap film in physicochemical equilibrium with a meniscus solution in an arbitrary thermodynamic state for which the mechanical equilibrium is supposed to be realized by compensating from the outside the force of thinning (or thickening) acting in the z-direction, i.~., it applies to an equilibrium film of arbitrary thickness formed at given vales of T, pi, ~1~.p’, paEssentially the same “D definition has frequently been applied in the past when evaluating =D from kinetic measurements on the thi;ming process which enable the determination af mf. When ht becomes largeenough, ITS= -Ap,and hence mD = 0. Thus it is clear that mD depends on the intei:iction between the two surfaces of tension which mechanically represent the soap film. In Fig. 6 the aD
*f
= -S’dT
-p’dVf
+ 2rfsdA
- nDAdht
+ 6
y@n:
(33)
I=1
Introducing the Gibbs free energy of the film this time as Gf’ = Ff the corresponding Gibbs free energy differential becomes dGf’
+p'Vf,
3
= -SfdT+
Vf&s'+2~f8dA-aDAti~t
+ C
(341
p@nf
i-1 Thus, the partial derivatives
(%yTp_ * . , A
ilGf’
dA T,p*,h.nf (-_)
mz
=
=
(35)
-”
2gs
(36)
provide aIternative definitions of fly and yf*. The integrated relation resulting from Fq. (33) is Gf* F' -= A A
+
prht
which leads to the subsequent muation of the Gibbs-Duhem _zd#"
=
dT-
htdp’
+ nDdht
+
3 C
i-l
kind
$dp,
But in this equation only four of the diffetintials dT, dp’, dpt (i = 1,2,3) ?re actually independent in view of the Gibbs-Duhem randition; Eq_ (16), that holds for the meniscus solulion. Making use of.Eq_‘(16) to eliminate dp,. from Eq. (38) we thus obtain the fundamental equation . -2d#”
=
Sf - h&)dT A_
+ nDdljt + i :.-
:
..
(39) ,_’
:
tt. follo~~from Eq. (39j that yfs and nb are interrelated accqrding to the .,.#+zp;.; _;’ -<- .. =::;._. :_- ., ;. -. :1 :- ..-_:_: : .! T,_-_‘_._-;. . :_z_-,;. . -_ .:_, I. ~, -! I ‘T_c-_: i__; -‘. ._.:;- ->-‘., :,-,::I
,_. _>=i, ,_. _.. .,..-. I ‘__, ,*+- :<_.., i_, .y
., _.
‘.
(41) where, in fact,thecanstraintofaconstantp’presaureisnotto
beneglected. The fundamental equation based on yfs with x2 and x3 as the composition variables, is the following as is moot readily verified by directly inserting the definitions of yfs and XE In Eq. (23).
-zS,dp’
(421
Although in the normal case, the contribution from the last 2+dp’ term in this equation is expected to be quite small it may still be of significance_ This. circumstance is exemplified in Figs. 5 and 7_ Figure 7 refers to a free equilib-. rium soslp film (xl = 0) where fat is varied by changing p’ whereas in Fig. 6, ht is varied by changing sr at constant T, x*, x3, p*;p,,; From Eq. (42): (43)
.
Thus -mD is twice the rate of change of 7% with ht at cqnstant state-of the meniscus sahtion as is hkewise_expressed, by-Eq. (41)‘*, : Uponcross-differentiating Eq. (42) we obtain ;_. . _. .’ aSv
a=23
(fp’ 1 _-
T.h.xa,X, .-;”
=-
( 1
_,
.
I
,._;
..
;-;
::
,._:
.
2-
aAtl:-. T.-xI.X,,p* , ;:. . . . :
I ‘.;_!-=
.
-.. :
!.
‘.
. ’ ._-
(44) _.
287
xxi
400
Jo0
6w
100
wo _-c--
‘coo
,‘c-u,
h;CA
_----c
//
/
_fl
rc-
-c--
Fig_7. Diagmm shotin the dependence of yc’ an he for CB soap fiIms of equilibrium thickness (SD = Ap) at constant 2’. p. and meniscus compoaitian but variable p’. The dashed paAt of the curves relale to negative Ap values. The same baaIc data have been wed as in Fig. I!_
268
Fig-S. A mechanical malogue of the soap CiIm system. ht may he varied by changing rf while simukaneo~ly adjusting the meniscus pi&on so ss to keepp’ con&ant. AItematively, l& can he varied keeping -f = 0 and changing p’. =p is represented by the force of a _ spring coupled to the surfaces af ten&n. I D is practicalty idependent of p’. i.e., r~ is a function of he only at con&ant T and meniscus composition. FUNDAMENTAL
EQUATIONS
ON DIFFERENCE
FORM
The experimental determination of the contact angle, 0, at the platiau border which separates the film from the meniscus, involves a direct compariI son- between the fitm surface tension rf!- and the surface tension of the meniscus, r” _-Methods are nof available which enable highly accurate measurements even of small (G 19) contact angles [ l&18]. At mechanical .equilibrium we obviously have rfs = y%os~ or :.
AT
.
7 ‘$6 -- r” .
f .r%Qit;g)
.._
..-
-1. : : .-.
. _
_::
-‘,_
__ : ...(45)
(46)
subscript (t; ¬ing surface excess relative to the surface of tension at t = h&2. Eq. (46) clearly agrees with the general form of the Gibbs surface tension equation valid for an interface of variable curvature insofar as the condition et =0 is fulfilled, U ext we wish to compare the real soap film with the idealized soap film formed under identical conditions and having the same thickness as the real film, i.e., ht = ht , ie_ wqabo presuppose that 6(,) = 6,,,+ implying that and rI = E‘,,ld. Introducing the notation ht,) i h(,&
(471 (i=
23)
etc., where, e.g., &, equals I’[ of a thick (hi) real film with nD = 9 diminished by (hi - &)c;, we can derive the following fundamental equation making use of Eq. (39) as the starting point:
-Zd(Ar)
= A
dT+nodht+
5 Al$dpi is2
With ht replaced by h<,) and written in ti less satisfactory form this equation has been~utilized previously by De Feijter et al. (2,361. It is presumably accurate.eno.ugh for most applications_ Even better suited for actual applicationa on coutact angle data is the correspondmg equation with T, ht, xa, xKj as the independent variables (cf. Eq. (42)): ,.
260
mined nD - ht functions recorded with T, x2, and 1c3as parameters. The relation between integral quantities a&ociated with Eqs. (48) and (49) is (cf. Eq_ (37)):
from erhich the free energy difference between the real and the idealized film is obtained. This equation was applied above when making the calculations for Figs. 5-7. Then Af’f , AI?: 70 was assumed to hold for the CB films considered and, moreover, we estimated the free energy difference A(F’/A ) from the expression (CGS units) [la] :
1tP2
- -+ 48,4,/c’; exp[-3.28.111’ 12mht2
fi&]
where the conventional terms are inserted for the free energy changes due to the van der W&s attraction and the doubfz layer’ repulsion_ For freesoap films of equilibrium thicI:ness studied at constant pa we have RD = Ap and ht = (T, x2. x3, Ap). Thus, when performing Ay measure me& at a fixed capillary pressure, Ap, and constant ambient pressure pa, the equilibrium thickness ht depends only on T, x2, x3. Under such circumstances contributions to the derivatives of Ay with-respect to T, x2, 1~~ will genemlly be obtained from_ the rf&ht term in Eq_ (49)_ For Newton black (MB) films, however, with ti typical equilibrium thickness of iat = 35 A 1171 these contributions appear. to be negligible since for such films ht is practically independent of T, x2. x3, and’dp. For common black (CB) soap films with I+ ranging between 50 and 10130A; the siruation is quit& different, Le., the equilibrium lhickuess is often strongly st&te dependent for these films. The data collected in Table 1 [14,X6] suggat that, as a rule, we can introduce the approximation dht (TV xKlrx3. Ap) = 0 for NB films whereas the Apdht term may well be a siieable term for CB films_’ .
--
CB film
CBfiIm’
NB film
hc=700A
he-200A
ht=35A
261 REMOVALPI?
THE CONSTRAINT
r’, = 0
The thermodynamic fundrrmental equations deduced in the previous sections include the simplifying assumption that -the film content of the gaseous component, is negtigiblu when the two surfaces of tension at z = Cht/Z are taken as the dividing surfaces which separate the film from the gas phase. For realistic “inert” gases such as He, Ar, N1, O2 this condition may not be futfilled with sufficient accuracy in view of the dubility of the gas in the! aqueous core of the film. It is hence of interea to investigate the consequences of rerrioving the constraint I$ = 0, Our starting point is then Eq. (15) or Eq. (38) supplemented with an additional rfdpl term, viz.,
rf ,
In order to transform these equations into forms corresponding to Eqs. (181 and (39) we make use of the Gibbs--Duhem conditions for the gas phase and the meniscus phase: ~“dT+c:‘d~, &dT+
eczdp,
&c;dpi-dp’ I=1
-dp, = 0
= 0
64) WI
with the purpose of eliminating the differenti& do, and dp+ The resulting equations are algebraidally c’o’niplex_However, if, as a more realistic approximation,‘Me assume that rf 7 ht,,c4, .- .' c: denoting the concentration ef the gasc&S’&&&ent ‘dissolved in the m&is&s ph& and, furthkinoce, take into’zicc&irit H&t ,thoq&ofi&t (C~c;)/(c;c~) at r&m titmperatuti’ typidally i$-ieti th&‘IbLd for~~utibga& ti those Ii&d &ov& it is kadily’dem&retrated tb&tthe’f&ida&e&l equai&;na’ (l&&d ($9) h&v&d ptiviouafy (on &+n&g thad Tf 2 (@-rcm&-&f&i&t[y &u&e for vir$&@dl a&&&ions;’
262
-26J&’
(57)
By assuming that I$ = hC,& = 0 (the
and that the partial derivatives
meniscus solution is very dilute in component 4) the
double sums in the above equations can be reduced as follows: .&
*c2 {r~-&m,ci)(~k)
*k
T,P’,Xl
=
k$$
i$2(ri--
mc
T,p’,q
hk
Hence, a close correspondance with Eqs. (23) and (42) is achieved. The importit& fundamental equation5 on difference form derived in the previous section, viz., E+. (48) and (49), remain valid in unchanged form even when rf #Cl as is most readily verified on the basis of the exact Eqs. (53) and (57). Thus by assuming that I’: =l$ id and rf = rf iti when ht id equals ht. Eq. (48) is generated from Eq. (5b).‘If, in addition: we assum;? that &v=sv,td and that the cross derivatkfes (a&ax& 4 x x and are equal to zero, Eq. (49) is oi$k&d&om Eq. (57). @f131aX4)T,p-.xCt,~, SUMMARY
AND
CONCLUSiONS
The thermodynamics of thin acup filma as developed in the present paper is based upon the,conception that. two weakly interacting fluid intorkcial
283
solid surfaces submersed in a fluid phase. Also, the explicit introduction of thermodynamic excess properties of the soap fiIm reIative to the meniscus solution is deliberately avoided in order to gain simplicity. As a meniscus phase in necessarily part of the soap film syakm we may chose either the pressure, p’, in this phase or the gas phase pressure pa as the reference pressure inside the fiim when defining the film tension and the disjoining pressure. Thus the overall film tensio;l, 7’. and the effective film disjoining pressure, rf, represent the deviations from pn in the directions parallel with and perpendicuk to the seap film, respectively, whereas the film surface tension, yb, and the Derjaguin disjoining pressure, ng, accounts for the corresponding deviations relative top’. The main rer;ults LB!this investigation are embodied in the various fundamental equations deduced - Ew. (15). (1% W), WV, (4% 1421, $421, and (49) - which zarebased on either of the quantities TV, ~~~~or Ay ‘r s - 7m and are formulated in terms of both@ and x varisbles. All of the fundamental equations obtained include novel features iargeXy resulting from our fundamentaUy different approach as compared with the previous thermodynamic treatments of thin soap films available. In order to survey which independent state variables the characteristic thermoriynamic film quantities actually may depend upon, Table 2 below has been prepared where ++ indicates a strong or moderate dependence and + a weak dependence. We have indicated the consequences of the fact that A i$ for the surfactant compnent generally is expected to be a very small quantity IW. The Dejaguin disjoining pressure sD appears to be a convenient quantity from a theoretical point of view as in a rather clearcut way it reElecta the magTABLE
2
If
-.
-T
++
Ir,fx*
u,fxs
++
++
~~ (fef)*
++
++
rgr p (feE)
++
++
++
++
P’
++
PO
AP
ht
+
++
++
+
++
+
++
Y
++
++ +
I
264
nitude of the long range interactions between the two faces of the soap film at least inrofar as the monolayer structuresat the film surfaces remain essentially unaffected by & changer. But still, in the thermudynamics of soap films, the quantity 4 y undoubtedly plnys a more important role. It is also evident that it is only needed to introduce a disjoining pressurewhen dealing with thin films if the films are of such a nature that ht actualIy exhibits some appreciable state dependence or when ht can be directly varied by external agencies. With fairly modest changed the formal description of thin soap films given here should be appticable also for dispersed emulsion droplets. When discussing the stability of such systems it is evidently the effective disjoining pressure nf rather than mD that is of primazy importance. Concerning the experhnental possibilities at hand for exploring the thermodynamic properties of thin equilibrium soap film, it is clear that contact angle measurements arc particukrly useful. The alternative procedure to integrate mD - hl isotherms requires much more data collection for obtaining the same amount of thermodynamic information. When an extremely precise method to determine the film tension yf ia avaihble, e.g. by optically measuring the curvature of the film at a minute pressure difference across the film, direct yf measurements may also prove to be useful (cf. Eq. (19)). ACKNOWLEDGEMENT IVe wish to express our gratitude
to Dr. J.A.
ments on a preliminary versian of this paper.
de
Fejter for valuable com-
REFERENCES 1 2
3 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18
B.V. loerjaguinand M. Kussakov, lzv. AN. S.S.S.R., Ser. Khim., 6 (1936) 741. J.A. de Feijter, J.B. RIjnbout and A. Vrij. J. ColbId EnterfaceHci., 64 (1978) 268. B.V_ Toshev and LB. Ivanov, ColMd and Polymer %i., 263 (1976) 658,693. A.I. Rusanov, in B.V. Derjaguh (Rd.), Research in Surface Forces, Vol. 3, ConsuItanb Bureau, Plenum Press, New York, p. 111,197l. B.V. Derjaguin and N.V. Chutaev, J. Colloid Interface Sci., -66 (1978) 389. A-I. Rusanov, PhasengleichgetichL und Grr?nz~chenergcheinungen, Akademie-Verfg, Berlin, Chapter 14,1978. F-0. Koenig, J. Chem. Phys., 16 (1950) 449. J.N. Israelachvili, D-J. Mitchell and B.W. Ninham, Trans. Faraday SOC. II, 72 (197&) 1626. T.L. Hill, J. Phys_ Chem., 66 (1962) 526. E.A. Guggenheim. Thermodynamics. 6th Ed., North-Holland, Amsterdam, p. 46,1977. S-G. Ash, D.H. Eweett and .C..F. Radke, %bans. Faraday Sot. II, 69 (1973) 1266. A. 3chebrdko, Adv. CcUoid Ynterfaace Sci., 1,(1967) 391. A. mumkin, Zhur. Piz. Khim., 12 (1938) 337. J.A. de Feijfer, Contact Angles in Soap FiIms, Thesis, Utrecht, 1973. J.S. Cbnie, J.F. Goodman and B-T. Ingram, in E. Matijevic’(Ed.), Surface ana Corn!!! .. Science, Vot. 3,WUey Inter6cIence, BeW York, 1971, pi 167. J.A. de Feijter and A. Vrij. J. Calloid Interface Sci, 64 (1978) 269. ; n_ LA. de Feijter and A. Vrij, J. C$Md_Intcrfacs S&nce, 70 (1979) 466. R. Buscall and R.H. Oitenwill, in D.H. Everett (Ed.), CoIlaid ,V@, 2, &m. Chem. . .. Science, .. . _ . s43c., 1976.