0 Copyright I988 American Chemical Society
The ACS Joumal of
Surfaces and Colloids MAY/ JUNE, 1988 VOLUME 4,NUMBER 3
Reviews Line Tension in Three-phase Equilibrium Systems B. V. Toshev, D. Platikanov, and A. Scheludko* Department of Physical Chemistry, University of Sofia, 1126 Sofia, Bulgaria Received August 5, 1987. I n Final Form: December 28, 1987 The thermodynamics and mechanics of three-phase contact line systems are developed. According to Gibbs, the line tension of the contact line is determined as an excess quantity which brings into correspondence two three-phasecontact systems: a real system with a transition region and an idealized system without it. &timaw of thisquantity are given for several cases including that of soap films,where long-range interaction forces play an essential role. The line tension can be negative or positive, and this is discussed in detail. From the 52 potential formulas for the work of nuclei formation in supersaturation systems are obtained. For heterogeneous nucleation, a negative line tension implies condensation without a barrier, and experimental data on the critical supersaturationfor nucleation of water on hexadecane are interpreted on that basis. These data are obtained by using the reverse Wilson chamber (RWC), which is described. The value of line tension thus obtained is -1.9 X lop6dyn. This value is close to the values of the line tension measured by other experimental techniques, which are also described. Measurements on Newton black film systems show that the line tension can change sign as the salt concentration is varied. 1. Introduction
Two-phase equilibrium is analyzed in detail in Gibbs’ thermodynamics of heterogeneous systems1by introducing excesses of energy, entropy, etc., per unit area of the interface. This is also the way of d e f i i g the surface tension u at the surface of discontinuity between the bulk phases. The latter is the source of the capillary pressure, the difference in pressure of the equilibrium bulk phases. According to Laplace’s formula, this is determined by the two main radii of curvature, R1 and R2,at any point of the surface of tension:
P, = u(;
+
);
This description is complete when a fluid phase surrounds another bulk phase. The application of these ideas to the case of a drop attached a t the interface between other bulk phases results in the Neumann-Young formula (ref 1,p 259). This analysis is obviously incomplete be-
I.
(1) Gibbs, J. W . The Scientific Papers; Dover: New York, 1961; Vol.
cause, besides the three surfaces of discontinuity, there also exists a line of discontinuity. Gibbs has commented on this fact in a rather brief footnote (ref 1,p 288). He noted that a complete description of contact line systems requires the introduction of excess terms for the energy, entropy, etc., per unit length of the line of discontinuity and also of line tension but did not use any mathematical formulation. The introduction of a line tension K leads to a formula analogous to eq 1 U, = K / r (2) where the two-dimensional pressure u,, acting toward the center of the contact perimeter, is determined by the radius of curvature, r, of the line of tension. It seems that eq 2 has been first introduced by Vesselovsky and Pertzov2 in a way similar to that used by Laplace for deriving eq 1. These authors also corrected the Neumann-Young equilibrium condition by adding the Q, term to the usual force balance for the system shown in Figure 1. U3 - U1 COS CY - U2 COS 0 - K/r = 0 (3) (2) Veseelovsky, V. S.; Pertzov, V. N. Zh.Fiz. Khim. 1936, 8, 246.
0143-1463/88/2~04-0489$01.50/0 0 1988 American Chemical Society
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490 Langmuir, Vol. 4, No. 3,1988
Figure 1. Fluid drop at a fluid interface in the absence of graviw ul, u2, u3, ox,two-dimensional forces; a
+ 8, the angle between
the drop's surfaces, r, the radius of the contact perimeter.
Figure 2. Liquid drop on a solid substrate: 0, the contact angle; R, the radius of curvature of the drop's surface; r, the radius of the contact perimeter.
The introduction of K in eq 3 is a fundamental change in the well-known Neumann-Young relationship (Figure 1) u3 - U' cos a, - u2 cos 0, = 0 (4) Similarly, for a solid phase with a plane surface (Figure 2) 6 3 - U2 - U1 COS 8, - K / P = 0 (3') which changes basically the classical expression u3 - u2
- u1 cos e,
=
o
(4')
Figure 3. Foam film in equilibrium with cylindrical meniscus. The subsystem with volume dV + dV8 = const in the form of a redangular pardelpiped with side dx in the plane of the drawing and a unit side in a direction perpendicular to the drawing and parallel to the straight contact line.
2. Line Tension: Definition and Estimates As already noted, according to Gibbs,' the line tension K must be defined as an excess quantity which puts into equality two systems: a real system with a three-phase transition contact zone and an indealized system without a three-phase transition contact zone. In the definition of both the surface tension u and of the line tension K , the 0 potential (constant temperature T,chemical potentials p, and total volume V of the thermodynamic system) is the extensive thermodynamic quantity which characterizes the system studied. (For the role of 0 = F - &;Nit where F is the Helmholtz free energy and Niare the total masses of the individual chemical components, in the thermodynamics of surfaces, see ref 1 and 4.) Let us consider a system with three plane surfaces with surface tensions ul, u2, and u3 which meet each other in a straight line of length 1. The angles between the surfaces are completely determined by the surface tensions; e.g., they are all 120° if u1 = u2 = u3. According to Gibbs, we shall consider two systems: a real one with variable surface tension in a region close to the contact line and an idealized one with constant surface tensions. Since the phase surfaces are plane, the pressures P in the corresponding bulk phases are equal, so that there are no volume terms in the expression for Qrd - Qidad of the system studied. Thus
The term Klr, added in eq 3, makes the contact angles a, 0,and 0 dependent on r ; a,, om,and 8, stand for the contact angles at r so that Klr 0. This "size- effect is determined by the value of K , and if K were extremely small in magnitude, then the effect would only appear when the droplet has molecular dimensions; therefore, there would be no practical meaning to the problem. For this reason, the evaluation of K is of considerable importance. This can be done either by statistical mechanical consideration of the molecular interactions in the three bulk phases and in the line contact region or by direct experimental measurement. There are now several independent ways of accomplishing the latter (see section 4). The thermodynamica and mechanics of the three-phase contact with some applications to the theory of flotation and of heterogeneous nucleation along with the description of some experimental techniques for measuring K were presented in our paper published in the memorial volume devoted to the centennial of Gibbs' theory of ~apillarity.~ However, essential progress in the experimental as well as the theoretical investigation of the line tension problem has been achieved in recent years. This up-to-date development is critically analyzed in the present paper.
where u ) are ~ the local, varying surface tensions which, far from the three-phase contact line, become equal to the constant surface tensions ui;Oi are the areas of the phase surfaces. Obviously ai are expediently determined boundaries of the three-phase contact zone through which B ~ the , corresponding integral excess quantities, are also defined. Both 6i and ai are determined by the nature of the phases included in the system and the interactions of the molecules of these phases in the bulk as well as in the contact region; they depend also on the geometry of the system studied. Since B~ may be negative, the line tension may also have a negative value. This difference between line tension and surface tension was specifically commented on by Gibbs (ref 1, p 296). By making use of eq 5, one may estimate the value of the line tension (for more details, see ref 5-7). Moreover, if we simply define K as the work of forming a contact line of unit length while u is the work of forming a surface of unit area, then, formally, the contact line has an "area" 16, so that K l ( Z 6 ) u. If we arbitrarily assume u = 50 dyn/cm
(3) Scheludko, A.; Toshev, B. V.; Platikanov, D. In The Modern Theory of Capillarity. The Centennial of Gibbs' Theory of Capillarity; Rusanov, A. I., Goodrich, F. C., Me.; Khimia: Leningrad, 1980. The Modern Theory of Capillarity. The Centennial of Gibbs' Theory of Capillarity; Goodrich, F. C., Rusanov, A. I., Sonntag, H., Bttlow, M., Eds.; Akademie-Verlag: Berlin, 1981. God. Sofii. Uniu., Prir. Mat.-Fak. 1976/1977, 71(1), 111.
(4) Boruvka, L.; Rotenberg, Y.; Neumann, A. W. Langmuir 1985,1, 40. (5) Kerins, J.; Widom, B. J . Chem. Phys. 1982, 77, 2061. (6)Rpwlinson, J. S.; Widom, B. Molecular Theory of Capillarity; University Press: Oxford, 1982. (7) Scheludko, A.; Toshev, B.V. Dokl. Bulg. Akad. Nauk. 1987,40(1), 75.
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-
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Langmuir, Vol. 4, No. 3,1988 491
and 6 = cm (molecular dimensions), then K = 5 X lo* dyn, which would seem to be too small a quantity to have practical significance? However, in some special cases, e.g., soap fiis with electrostatic double-layer interactions, one may expect 6 > cm.Thus for simple systems (e.g., Lennard-Jones fluids), one would expect that 1. < lo* dyn? but for more complicated systems (e.g., foam films), the magnitude of K is expected to be larger. Recently such an opinion has been expressed also by Rowlinson.lo Next we apply the same Gibbs’ recipe to the case shown in Figure 3 of a plane-parallel foam film f of thickness h in equilibrium with a gas phase g (pressure pg) and with a bulk liquid phase 1 in the form of a cylinder so that a straight contact line is formed. In the absence of gravity, the capillary pressure is P, = pg - p’ = d/R, where p’ is the pressure of the bulk liquid. Then
“I
I
I
b)
Figure 4. (a) Typical isotherm of the disjoining pressure n(2); (b) isotherm of u(Z) obtained from aaff/a2 = -n.
as is well-known, the maximum in Figwe 4a decreases with
increasing electrolyte concentration of the solution from which the film is formed. This implies the possibility of changing the sign of the line tension from the less typical The left-hand side of eq 6 is the s2 potential of a real positive to the more expected (according to the theory) subsystem, shown in Figure 3, and the terms denoted by negative. This th0ughtlS which has not been stated in ref (id) in the right-hand side of eq 6 define the s2 potential 12 is in accordance with the experimental data. of the same subsystem after proper idealization. The The presence of a liquid film in the system seems to be straight contact line in the idealized system is a result of an essential feature of the theory of de Feijter and Vrij. an extrapolation of the surfaces of the film (separated by The transition from a liquid film of finite thickness to an Z)at constant uf and the meniscus at constant capillary adsorption layer or a single interface (h 0) cannot be pressure with constant 6’. Then, it immediately follows evaluated because of the lack of knowledge about the infrom eq 6, with dV = Zdx, dVid)= Z(id)dx,dO = dx/cos teractions a t very short distances. Thus the relation be8, and docid)= dx/cos (Figure 3),” that tween complex capillary systems with long-range interactions and simple systems with short-range interactions seems to be still an open question. The theories presented here are entirely based on thermodynamics. Models must be used when applying Equation 7 is de Feijter and Vrij’s formula obtained earlier these theories, and there are some indications that the value of the line tension has a certain sensitivity to the by the force-balance method.12 model used for the ~alculation.”~Furthermore, there are We emphasize again that one has to distinguish between u, 8, and Z(x) in the two systems. For instance u(Z) # did) other general questions. For instance, one might admit an ostensible arbitrariness in the division into volumes and = d. Here d is the surface tension of the liquid from which the film is formed, far from the contact with it. In acsurfaces according to the theory of de Feijter and Vrij (see cordance with Derjaguin’s plane-parallel approximation,13 eq 6).16 However, the molecular theory of the line tension u(Z) = d + ‘/JZ II dZ and uf = u’ + II dZ. Acmay be developed by the methods of statistical mechanics cording to eq 7, one may presume that the line tension and hydrostatics as well.17J8 Unfortunately, the results exists because of the long-range interaction forces in the obtained are too complicated and, as a rule, do not lead transition region between the film and the bulk liquid. to exact solutions. This situation seems to be similar to Then, if the disjoining pressure isotherm II(Z) is available, that of the two-phase equilibrium, where exact solutions for the surface tension are available only in some special one may easily calculate K . Such calculations with some disjoining pressure isotherms have been carried O U ~ and ~ ~ J ~ cases. dyn. The line tension for free soap films yield 1.1 < 3. Energetics of the Phase Boundaries has the lower values.12 The data of ref 14 refer to the case of a wetting film on a solid surface, and the calculations In section 1we discussed briefly the force-balance meare carried out with the assumption that the disjoining thod for obtaining the conditions of mechanical equilibpressure in the transition zone between the film and the rium of the capillary system studied. The minimization bulk liquid affects the profile of the surface of that zone of the Q potential at constant temperature, chemical powithout changing the surface tension there. tentials, and volume is the thermodynamic way of deriving Figure 4a shows a typical isotherm of the disjoining those conditions. pressure II(Z), and Figure 4b shows u(Z) obtained thereSome comments are needed about the equivalence of the from by making use of the approximation a2u/dZ = -II, thermodynamic and force-balance approaches, which both noted above. According to the formula K = Jhm(usin 8 give equations of the type of Neumann-Young’s relationd sin dZ,12which is the alternative to eq 7, one may ship. As is known, for the case of a solid, one has to expect a positive value of the line tension K. Furthermore, distinguish between the process of stretching the phase boundaries and that of forming them (ref 1, p 315). Thermodynamic analysislg shows that equations of the (8)Pethica, B. A. J. Colloid Interface Sci. 1977,62, 567. P,dV
+ 2ud0 = P,dVd) + 2u(id)dO(id)+ dK
(6)
-
l/pJr
(9) Buff, F. P.; Saltsburg, H. J. Chem. Phys. 1957, 26, 23. Faraday Trans. 2 1983, 79, 77. (10)Rowlinson, J. 5. J. Chem. SOC. (11)Toshev, B. V. Dokl. Bulg. Akad. Nauk. 1986, 39(10),87. (12)de Feijter, J. A.; Vrij, A. J. Electroanal. Chem. 1972, 37, 9. (13)Derjaguin, B. V. Kolloidn. Zh. 1955, 17, 207. Derjaguin, B. V.; Churaev, N. V. Kolloidn. Zh. 1976, 38,438. (14)Churaev, N.V.; Starov, V. M.; Derjaguin, B. V. J. Colloid Interface Sci. 1982, 89, 16.
(15)Platikanov,D.; Nedyalkov, M.; Naateva, V. God. Sofii. Uniu.,A i r . Mat.-Fak. 1979, 73(2), 9. (16)-Eriksson,J. C.; Toshev, B. V. Colloid Polym. Sci. 1986,264,807. (17)Tarazona, P.;Navascu&, G. J. Chem. Phys. 1981, 75, 3114. (18)Vignes-Adler, M.;Brenner, H. J. Colloid Interface Sci. 1985,103, 11.
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492 Langmuir, Vol. 4, No. 3, 1988
equilibrium nucleus having a spherical shape, and the work of its formation W = (Q" - fl')/z is w = y3uo - [(PIU - Pl')Vt]/Z (10) If we consider the nuclei as a second "perfect gas" component so that PznV= PtVt + zkT, then a generalization of the well-known Gibbs' formula, W = '/3u0 = '/Zpav(ref 1, p 258), immediately follows from eq 10:
(5,
Figure 5. Sketch of a heavy lens attached at a fluid interface: @ and 9,angles of Neumann's triangle; the angle J. depends on gravity.
Neumann-Young type are of energetical meaning and that their force interpretation is possible only for open fluid systems. If there exists a solid in the system studied, then the force-balance involves either adding terms of Shuttleworth's type or keeping the Neumann-Y oung-type conditions of mechanical equilibrium but with the "superficial tensions of the fluid in contact with a solid" introduced by Gibbs (ref 1,p 329). Nevertheless, we shall use the symbols u also in the case of a drop on a solid substrate (Young's rule), but we shall always remember the difference between the quantities involved in the Neumann-Young equation for this case and those in the case of all-fluid systems with contact angles. For the sake of further analysis, it is worth examining the behavior of a heavy drop attached at the boundary between two fluids (Figure 5). According to the largescale experimental and theoretical investigation of Lyons,20in the case of heavy lenses, the Neumann-Young rule is affected by gravitation and, hence, is not valid. However, a thermodynamic analysisz1carried out by minimization of the fl potential of the system studied, 6Q = 6(flv + fl, flp~M3 = 0, does not confirm this statement. It shows that gravity deforms the phase surfaces, nevertheless, the Neumann-Young equation holds; i.e., the influence of gravitation reduces to a rotation of the Neumann triangle around an axis perpendicular to the plane of the sketch (Figure 5 ) . In order to take into account the line energy one adds fl, = 2 a r ~to the expression of fl, and the final result is as follows: 6 3 = 6 2 cos (p - l)) + u1 cos (a + l)) + ( K / T ) cos l) = -uz cos Q - u1 cos @ + ( K / r ) cos l), u1 sin a u3 sin l) = uz sin @ (8)
+
+
As it is known that (Afl)T,vr has the meaning of reversible work, it provides a simpie way of deriving the work, W, of formation of a nucleus in supersaturated systems. The nucleus is a small but macroscopic particle of the new phase, which is in unstable equilibrium with its surroundings. The probability of formation of a different fluid phase within any homogeneous fluid is proportional to exp(-W/kr), where k is Boltzmann's constant and W = Afl when this transition occurs at constant supersaturation. So, for the homogeneous condensation in supersaturated vapors we writez2 fl' = -P,'vt = -P1"(Vt - v) - P2"V
+ UO
(9) Here Vt is the whole volume of the system and, a t condensation, z nuclei are formed with a volume V = zu and area 0 = 20. P211- PtI is the capillary pressure of the flI1
(19)Eriksson, J. C. Acta Chem. S c a d . 1977,A31,235. Toshev, B.V.; Eriksson, J. C . God. Sojii. Uniu., Prir. Mat.-Fak. 197511976,70(1), 75. (20)Lyons, B.G . J. Chem. SOC.1930,623. (21)Toshev, B. V. Dokl. Bulg. Akad. Nauk. 1980, 33(8), 1095. (22)Radoev, B.;Scheludko, A.; Toshev, B.V. J. Colloid Interface Sci. 1986,113,1.
W = KUO- kT
f/ZPav- kT
(10')
This result disagrees with the treatment of Lothe and Pound,B which includes statistically calculated corrections to Gibbs' formula leading to a lO"-fold increase in nucleation frequency. According to our formula, this increase is only 2.7 times; consequently, the effect would not be manifested experimentally. In the case of heterogeneous nucleation, the line energy of the nucleus contact perimeter, I, has to be taken into account (Figure 1). Then the described procedure with the conditions of mechanical equilibrium yieldsz4 = f/z(PaU K I ) (11)
w
+
This result holds for nuclei immobilized on the surface where they have formed. Different cases of nuclei formation are accounted for in eq 11: the classical formula for homogeneous as well as heterogeneous nucleation follows directly from eq 11with K = 0; for two-dimensional condensation u = 0 and W = l / Z K 1 . For the case of condensation on a solid substrate eq 11 in a more complicated form has been first obtained by Gret~.~~ The vapor pressure p of a small liquid drop is determined by the Thomson-Gibbs equation P 2uD kTln-=PR where b is the molecular volume. When the drop is formed on a substrate and the line energy of the contact perimeter is taken into account, the capillary pressure of the drop, 2u/R, is not affected. However, in contrast to the case of homogeneous nucleation, where all the values between 0 and of R, the radius of curvature of tbe drop, are allowed; now, the values of R are restricted by the condition of tangential mechanical equilibrium. This requires, for the case of a drop on a plane surface (Figure 2 with ul u and r = R sin e), that
-K / -U - (COS e, R
- COS e) sin e = R
(13)
Thus, neglecting gravitation (for small drops gravity is not important), we have P 2uZD 2 2 6R kT In - = -(cos 8, - cos e) sin 0 = - (14) P- kTK k TK Thus the vapor preesure of a droplet on a substrate depends on the contact angle 0 and has a maximum when dK/dB = 0. This corresponds to an angle Om given by
e,,* = y4 COS e, (1 * (1 + 8/cos2 e,)1/2) (15) with two roots: Om, < 8, at K < 0; em > 0, at K > 0. For a given system 6 determines r teq 3 9 , and , as the COS
two determine the volume of the droplet and thus the number n of molecules in it, the vapor pressure has also (23)Lothe, J.; Pound, G . J. Chem. Phys. 1961,36,2080. (24)Scheludko, A.;Chakarov, V.; Toshev, B. V. J . Colloid Interface Sci. 1981,82, 83. (25)Gretz, R.D.J. Chem. Phys. 1966,45,3160;Surf. Sci. 1966,5,239.
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Langmuir, Vol. 4, No. 3, 1988 493
a
l2ssssJ
i
Id
=I I
51d
IO3
51d Id
51d Id
510’
bl
\ t
I / , Id 51d Id
\ \2
51d Id.
--
51d Id
system the work of nucleus formation has to be expressed by
w = w-t - wst = l/z[p,(u~nst - U s s + K(l,,t I
\
Figure 7. Liquid nucleus partially wetting a spherical particle (of radius Ro)of foreign matter.
I 510’
n
Figure 6. Dependence of In (pip,) on the number of the molecules in a water droplet formed on a solid substrate: curve l, 8, = 180’ and K = 0; curve 2,8, = 45’ and K = 0; curve 3 , 8 , = 45’ and K = dyn; curve 4,8,45O and K = -106 dyn. Not the change in the vertical scale between a and b.
a maximum as a function of n. As an illustration, we show the dependence of In ( p / p , ) on n for a positive K = dyn (Figure 6, curve 3) and dyn (Figure 6, curve 4). The for a negative K = calculations were carried out for water with D = 2.93 X lWB cm3, u = 72 dyn/cm, kT = 4 X erg, and 8, = 4 5 O (according to eq 15 ,0 = 25’ and dm = 124O). For comparison, we show also the “uncorrecte&’ curves 1and 2 for K = 0. Curve 1 is for 8 = 180°, i.e., an entirely inactivated substrate, and corresponds to homogeneous nucleation. Curve 2 is for e = 45O (the same as for curves 3 and 4) and conforms to the classical theory of heterogeneous phase formation. It must be emphasized that the corrected curve of In ( p / p , ) vs n lies between the uncorrected curves for homogeneous and heterogeneous phase formation when K > 0, whereas for K < 0 it lies below those curves. The curve of In (pip,) vs n a t K > 0 is “noose” shaped (Figure 6, curve 3). At given n there exist two drops with different vapor pressure. The drop with the higher vapor pressure (upper branch) possesses a smaller radius of the contact perimeter and a larger contact angle. Thus, for the upper branch, the substrate is less active while, for the lower branch, it is more active. At supersaturations higher than that corresponding to the “tip” of the “noose”,vapor condensation obeys the theory of homogeneous phase formation. At supersaturations lower than that point, two drops of different size are in unstable (unst) equilibrium with their surroundings, and, obviously, the smaller one will be the drop nucleus; the work of its formation is given by e q 11. For K < 0, the dependence o,f In ( p l p , ) on n exhibits a maximum corresponding to K, with 0 = Om, (Figure 6, curve 4). At supersaturations lower than that corresponding to the maximum, drops of two different sizes are in equilibrium, just as in K > 0. What is different is that the smaller drops are in stable (st) equilibrium with the surroundings. This means that these drops are not nuclei. They belong to the initial state of the system a t any suIn such a persaturation In @ / p , ) < In (P/pJlatRIR,.
- I S ~ ] (16)
Obviously eq 16 is not burdened by the correction term kT (eq 10’). This case seems to be similar to that of condensation upon electrically charged free droplets.26 The drop nucleus in the recently reformulated classical theory of homogeneous nucleation with size-dependent surface tension of the nucleus behaves in a similar way.27 At Wst = Wmt (which corresponds to the maximum of the curve of vapor pressure vs drop size), condensation is barrierless. This implies the existence of a certain absolute critical supersaturation. An advantage of such an interpretation is that it avoids the necessity of determining the preexponential factor in Volmer’s kinetic equation. Then K
= (2R,,u2D)/[(kT In
( ~ / ~ m ) ) ] a t z = ~ ~ , (17)
with
Rm1= (COS e, - COS e m j sin e,, Let us point out that the first interpretations of experimental data on heterogeneous nucleation taking the line tension into account did not make use of eq 16 and 17.28*24129 Later the analysisg0followed the picture presented here in detail, with some generalization to the case of a drop settled on a liquid ~ u b s t r a t e . ~ ~ The theory of forming liquid nuclei on partially wetted spherical particle^^^*^^ has also been reformulated, taking into account the effect of a negative line tension.34 In this case (Figure 7) the tangential condition of mechanical equilibrium, eq 4,includes an additional term -(K/r) cos T . This result was obtained earlier by a thermodynamic analysis.36 The relationship between the angles 7 and 0 is (Figure 7) tan?=
sin 0 a - cos 6
-
where a = Ro/R. If the particles of foreign matter are sufficiently big, then a > 20/30,7 0, and the theory with (26) Tohmfor, G.; Volmer, M. Ann. Phys. 1938, 33, 109. (27) Raamuesen, D. H. J. Cryst. Growth 1982, 56, 45. (28) Scheludko, A. Colloids Surf. 1980,1, 191. (29) Navaacuh, G.; Mederos, L. Surf. Technol. 1982,17,79. (30) Scheludko, A. Colloids Surf. 1983, 7, 81. (31) Chakarov, V.; Scheludko, k; Zembala, M. J. Colloid Interface Sci. 1983,92,35. Chakarov, V. Colloid Polym. Sci. 1983,261,452. Scheludko, A.; Chakarov, V. Colloid Polym. Sci. 1983,261, 776. (32) Fletcher, N. H. J. Chem. Phys. 1958,29, 3. (33) Kraatanov, L.; Miloshev, G. Theoretical Basis of Phase Tramition of Water in the Atmosphere; Bulgarian Academie of Sciences: Sofis, 1976 (in Bulgarian with summary in Engliih and Russian). (34) Scheludko, A. D. J. Colloid Interface Sci. 1985,104, 471. (35) Toshev, B. V. God. Sofii. Uniu.,A i r . Mat.-Fak. 1974/1975,69(1), 25.
494 Langmuir, Vol. 4, No. 3, 1988
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negative line tension, described above, could be applied in these cases as well. It is worth mentioning that the effect of the line tension on the contact angle depends inherently on the geometry of the capillary system considered. At r = ?r/2 (e.g., wetting of an upright cylinder partially immersed in a liquid), the line tension does not affect the contact angle a t any radius of the cylinder. 4. Experimental Determination of the Line
Tension One of the possible ways of measuring the line tension ensues directly from the Neumann-Young equations8 in a form including the two-dimensional capillary pressure K / F . These equations require, at constant surface-and line tensions, a dependence of the contact angles on the contact line radius r. The experimental investigation of these dependencies at known ui values allows one to determine the value of K. Substantial difficulties spring from the very low value of K , which requires measurements at very small r. As a rule, in the experimentally feasible cases the term K I F is very small, and this lowers the accuracy of the measurements. Besides, hysteresis of the contact angle on solid surfaces can easily mask the small effect of the line tension. The study of the dependence of the contact angle 8 upon r has been initiated by Vesselevsky and Pertzov2 with a system consisting of a small gas bubble attached to a plane solid surface in an aqueous solution. As indicated in section 1, these authors introduced the term Klr into Young’s equation (4’). They have found experimentally a substantial decrease of 8 upon decreasing r and have calculated large positive values for K: from 0.13 to 1.4 dyn. Analogous results, by the same method but with a reverse system (small liquid drop attached to a plane solid surface in a gas medium), were obtained 43 years later by Good ~ saw a substantial decrease of 8 upon deand K o o , ~who creasing r and large negative values of K: from -0.58 to -1.72 dyn. In both investigations, bubbles (droplets) with contact radii of the order of magnitude of millimeters and contact angles of the order of tens of degrees have been studied. Although the substances used were different, the fact that in both cases a decrease in 8 upon decreasing r is observed (this decrease of 8 is much greater for the receding than for the advancing angles) as well as the very large absolute values of K obtained indicate that the observed 8(r) dependence is not due to the line tension K but is an effect of nonhomogenity of the solid surface. The author’s own comments contain the same interpretation, and in ref 36 the phenomenon is named “pseudo-line tension”. In systems consisting of fluid phases only, the Neumann-Young equations (8) can be used to determine the line tension by measuring the radius r of the contact line, the contact angles, and the three surface tensions for small droplets at a fluid interface. The first experiments of LangmuiP with floating oil lenses on water have not given results; later analysisB has shown that the observed effects were not due to line tension. Very interesting is the next attempt of Gershfeld and G0od,B9who have observed the three-phase system: a droplet of mineral oil (diameter 10-30 pm) partially penetrating the membrane of an echinoderm egg immersed in bulk water. All three interfacial tensions are very low, between 0.1 and 1.0 dyn/ (36) Good, R.J.; Koo, M. N. J. Colloid Interface Sci. 1979, 71, 283 (37) Langmuir, I. J. Chem. Phys. 1933, 1, 756. (38) Pujado, P. R.; Scriven, L. E. J. Colloid Interface Sei. 1972,40,82. (39) Gershfeld, N.L.; Good, R. J. J. Theor. BioE. 1967, 17, 246.
Figure 8. Small floating bubble with a Newton black film on its top.
cm. The geometric configuration of the surfaces indicated a positive line tension: K = 5 X lo4 dyn. A similar system, droplet dyads of a silicone-oil (1)and of aniline (2) in an aqueous medium (3), were studied by Torza and Mason.40 The radii of the droplets varied between 10 and 80 pm, and the interfacial tensions equaled a few dynes per centimeter. The photographs of five such dyads were analyzed, and the interfacial tension of anilinelwater, a=, and the line tension, K , have been calculated (uI2and 413 were measured separately). The values obtained for both 623 and K are rather scattered. In four of the cases K was found to be positive and in one case negative; the absolute values varying from 6 X lo4 to 6 X dyn. The authors point that such a large scatter reflects the inaccuracy of their measurements and that pure systems in which all three interfacial tensions can be measured exactly in separate experiments are necessary to obtain more reliable results. A floating dodecanol lens on a water surface was studied by Del Cerro and Jameson,4l who measured the radius of curvature of the lens surface by the method of differential shearing interferometry and established a decrease of the contact angle between the surfaces of the lens when its radius decreased. This result denotes the presence of a negative line tension, but the published data are few and do not allow one to accept as reliable the K values found to be of the order of magnitude of dyn. The authors themselves think that the effect is not due to the line tension.4I The experiments with floating dodecanol lenses on water surface have been repeated42by the same differential interferometric method, and the contact angle was found to be independent on r; i.e., no effect of line tension has been detected, but the droplets were relatively large. Another suitable system with fluid interfaces is the small circular black foam film (separating two gas phases) in contact with the bulk phase of the aqueous solution from which the film is formed. The presence of a line tension K along the contact line black film/bulk liquid can be expected on theoretical grounds12 and on the basis of many experimental observations going back to B0ys.4~ Attributing K to the increase of the surface in the transition zone black filmlthick (common) film, Boys estimated its value to be less than 5.25 X dyn in the case of the aqueous solution of sodium oleate and glycerol studied by him. Detailed investigations of the line tension in black foam f i i from sodium dodecyl sulfate (NaDoS) solutions were carried out by three different experimental methods in our laboratory. (40) Tom,S.; Mason, S. G. Kolloid 2. 2. Polym. 1971, 246, 593. (41) Del Cerro, M. C. G.; Jameson, G. J. Wetting, Spreading and Adhesion; Padday, J. F., Ed.; Academic: London, 1978; p 61. (42) Mingins, J.; Nikolov, A. D. God. Sofii. Univ., Prir. Mat.-Fak. 1981, 75, 3. (43)Boys, C. V. R o c . R. SOC.London, Ser. A 1912, 87A, 340.
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Langmuir, Vol. 4, No. 3, 1988 495
l 2 l II 2
11
l9 4
~
50
100
150
Rb,/*m
Figure 9. Plot of contact angle 0 vs bubble radius Rb: curve 1, 0.32 M NaC1, one bubble; curve 2, 0.45 M NaCl, two bubbles. The variation of the contact angle as a function of its radius can be conveniently studied for the system shown in Figure 8. A small bubble at the solution surface is in contact with the bulk gas phase through a small circular black film. The method of the “diminishing bubble” was developed to study this This method utilizes the fact that the gas in the small bubble is under higher pressure; it spontaneously passes through the black film into the bulk gas phase, and the bubble gradually decreases in size. The bubble is observed through two coaxial microscopes: in reflected light from above and in transmitted light from below (through the flat glass cell bottom). This allows the simultaneous photographing from both sides; thus at a given instant both the contact line radius r and the bubble radius Rb (Figure 8) are determined. From these data, with the aid of suitable formulas,45’46 the value of the contact angle 6’ is calculated for different bubble radii. In Figure 9 such results, obtained in ref 45, are presented. It is of particular interest to point out that, depending on the composition of the solution, both cases are observed: 6 decreasing with decreasing Rb (and r ) , as shown by curve 1 for 0.32 M NaC1, as well as increasing 8 with decreasing Rb, as shown by curve 2 for 0.45 M NaC1. The former shows a positive K , the latter a negative one. At an intermediate NaCl concentration (0.36 M) 8 is almost independent of Rb (and r); the effect of line tension is hardly noticeable because of its low value.44 The 6 vs Rb data allow estimation of K from each pair of measured Rb and r by making use of eq 8, rewritten here (Figure 8) cos (e + +) + K/r = cos (28 $1 + COS
+ + y sin (e + +) = u sin (28 + +) + u sin +
(8)
The solution surface tension u is measured separately, and the film tension y = 2u cos 8, is calculated with the value of the contact angle 8, obtained from the horizontal portion of the curve (Figure 9). Equation 19, obtained in ref 44 and 45 from eq 8 for small bubbles (e >> +), offers another possibility for determining K from e / r data: - =1 - - - -2U K 1 (19) COS e y y r The mean value of K of all separate measurements is calculated from the slope of the straight line plot of (cos e)-1 vs r-l. This approach was used in ref 44 for the statistical treatment of a large number of data obtained with several (44) Platikanov, D.; Nedyalkov, M.; Nasteva, V. J . Colloid Interface Sci. 1980, 75, 620. (45) Nedyalkov, M.; Platikanov, D. Abhandndlungen Akad. Wiss. DDR, VI Intern. Tag. Grenzfl. Stoffe; Akademie-Verlag: Berlin, 1986; No. lN, p 123. (46) Dimitrov, D. S. Dokl. Bulg. Akad. Nauk. 1977, 30, 269.
-C k C , , Figure 10. Plot of line tension
moc/ml’
of Newton black films and sodium dodecyl sulfate solutions vs NaCl concentration as determined by the “diminishingbubble method”. K
bubbles for each solution; the scattering of the results was greater there than in Figure 8, which reflects the experimental improvements reported in ref 45. In ref 44, we studied the whole range of NaCl concentrations in which Newton black films could be formed from a 0.05% NaDoS solution. A very interesting dependence of K on the electrolyte concentration was obtained (Figure 10). The absolute values of K obtained are less than about lo4 dyn, but they are positive at lower and negative at higher NaCl concentrations. These results can be interpreted on the basis of the theory presented in section 2 (eq 7; see also ref 12, 14, and 15). The method of the “diminishing bubblevu was used in ref 47 to measure K , but there, besides Rb and r, the radius of curvature Rf of the black film was measured also by the method of differential interferometry, as in ref 41. The purpose of this additional measurement was to determine simultaneously the film tension y, presuming its dependence on Rfi In ref 47 an inexplicably strong dependence of y on Rf is reported (y > 2u for a large part of the Rf range). This results in a highly anomalous dependence of K on r (varying between and dyn for r from 5 to 50 pm). These results are analyzed in detail in ref 48 on the basis of precise measurements of the film tension y over a wide range of Rf (70-700 pm) by a new meth0d.4~ The lack of any dependence of y on Rf, thus established over the entire Rf range, indicates that the values of y are incorrectly determined in ref 47, probably because of wrongly measured Rfi Therefore, the anomalous dependence of K on r reported in ref 47 is apparently erroneous. Another method by which the dependence of 6 on r may be studied for Newton black films from the same solutions is developed in ref 50. The films are formed in the center of a double-concave drop formed in the cylindrical hole of a sintered glass plate.61 The values of 6 as a function of r are determined in two ways: by differential interferometry in transmitted light and by measuring the capillary pressure in the meniscus. The same two solutions for which the data are already presented in Figure 9 have been studied. By this method again a decrease of 6 upon ~~
~~~~
(47) Kralchevsky, P. A.; Nikolov, A. D.; Ivanov, I. B. J. Colloid Interface Sci. 1986, 112, 132. (48) Platikanov, D.; Nedyalkov, M.; Scheludko, A,; Toshev, B. V. J. Colloid Interface Sci. 1988, 121, 100. (49) Platikanov.. D.:. Nedvalkov, M.; Rangelova, N. Colloid Polym. Sci. 1987, 265, 12. (50) Zorin, Z. M.; Platikanov, D.; Rangelova, N.; Scheludko,A. Surface Forces and Interface Liouid Lavers:. Deriaauin, _ - . B. V.,. Ed.;. Nauka: Moscow, 1983; p 200 (in Russian): (51) Exerowa, D.; Scheludko, A. Dokl. Bulg. Akad. Nauk. 1971,24,47.
496 Langmuir, Vol. 4, No. 3, 1988
-
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2Rbrm
Figure 11. Distribution in percent of unattached (open circles) and attached (solid circles) particles vs particle diameter 2Rb.
decreasing r (positive K ) at 0.32 M NaCl and an increase of 0 upon decreasing r (negative K ) at 0.45 M NaCl have been measured, but the absolute values of K are slightly higher than those determined in ref 44. The same method has been used by Kolarov and k i n s 2 to measure the line tension in common black films from an aqueous solution of NaDoS a t the same surfactant concentration but at lower concentration of NaCl(O.l M). The contact angles in this case are substantially smaller and are determined from the usual interferometric picture in reflected light by the so-called “topographic method”.s3 Increasing contact angles upon decreasing film radius were observed, leading to a negative line tension of -1.7 X lo* dyn, which agrees with the theoretical estimates for these relatively thick films.12 The topographic method has recently been used to study the transition zone between a wetting aqueous film on a silica surface and the adjacent meniscus.54 In the cases of complete wetting, when the extrapolated profile of the meniscus does not reach the plane film but passes at a minimum distance Ho from the solid surface, the quantity Hois a thermodynamic characteristic replacing the contact angle. Analogously to the latter, Ho also depends on r for small films, and in ref 54 an estimate of the line tension is carried out from the experimental data for pure water on silica: ita value is positive and does not exceed lod dyn. All the methods of determination of the line tension K mentioned so far make use of the variation of the contact angles a t small contact radii r. A different possibility of measuring K in the positive range of its values is offered by the phenomenon of formation of a new interface when a spherical particle (solid, liquid, or gaseous) of very small size is pressed into an interface dividing two bulk phases. The nascency and expansion of the contact (new interface) between the particle and the bulk phase are hindered by a force barrier due to the positive line tension. At the outset, when r is very small, the two-dimensional pressure K/r is large. An external force, pressing the particle into the surface (usually this is gravity), is necessary to overcome this force barrier. The external force must be larger than a given critical value, which depends on the value of K. On this basis two methods have been developed: in one, small glass spheres, in the other, small gas bubbles are pressed through a liquid phase into the liquid/gas interface. In the first methodss small glass spheres (balotini) of different sizes are treated with M aqueous octyltrimethylammonium bromide, and the wetting angles are
Figure 12. Schematic of the experimental cell of the “critical bubble method”: E, microelectrode; G, pulse generator; M, microscope.
about 25O. Then they are allowed to settle through a pendant drop of the solution. The glass spheres settled on the lower solution/gas interface are observed and photographed through a metalographic microscope in reflected light. The balotini radii Rb and the contact radii r (if the contact glass/air is formed) can be measured from the photographs. In Figure 11the distribution of particles having contact with air (solid circles) and those without such contact (open circles) is presented as a function of their diameter. Obviously, a new interface is formed predominantly by the larger balotini, whose weight is sufficient to overcome the line tension barrier. From these data the mean value of R,, the critical radius of the spheres below which contact cannot be formed, is determined. According to the analysis made in ref 3,55, and 56, this value is related to line tension through the expression K
= (2/3pgu)1/2(1 - COS 0,)R:
(20)
dyn For the w e studied in ref 55, the value of 1.25 X is thus obtained for K. The second method, based on the same principle, is described in ref 57. The measuring cell is shown in Figure 12. The small gas bubbles are generated by electrolysis on the microelectrode E and the required size of the bubble is attained by short rectangular dc pulses of suitable voltage. After a short time interval for saturation of the adsorption layer of the bubble, the latter is released by a high-voltage spike of negligible duration and allowed to rise to the surface, where it is observed through the microscope M. If, when it reaches the interface, the buoyancy force is sufficiently large (i.e., sufficiently large Rb) to overcome the force barrier due to the positive l i e tension, then a new contact surface, a Newton black film, forms. Otherwise (small Rb) the bubble remains beneath the surface, touching it but without forming a Newton black film. By microscopic measurements of Rb of many bubbles of different sizes, the critical radius R, is determined statistically. Above this critical radius, the line tension barrier can be overcome. In ref 57, the approximate treatment of the small bubbles as undeformable spheres yielded an expression analogous to eq 20. This expression was used to determine the value of K ; these values turned out to be very close to those found by the method of the diminishing bubble.& When the small deformation of the bubble by gravity was taken into account, the following expression was obtained:
(52)Kolarov, T.;Zorin, Z. M. Colloid Polym. Sei. 1979, 257, 1292. (53)Kolarov, T.;Scheludko, A.; Exerowa, D. T r a m . Faraday SOC.
1968, 64, 2864.
(54)Zorin, Z.M.;Platikanov, D.; Kolarov, T. Colloids Surf.1987,22, 147. (55)Mingins, J.;Scheludko, A. J.Chem. SOC.,Faraday Trans. 1 1979, 75, 1.
(56)Scheludko, A.; Toshev, B. V.; Boyadjiev, D. T. J. Chem. SOC., Faraday Trans. 1 1976, 72, 2815. (57)Platikanov, D.; Nedjalkov, M.; Scheludko, A. J. Colloid Interface Sci. 1980, 75,612.
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Langmuir, Vol. 4, No. 3, 1988 497 I
1
I
I
1
I
1
1
n
13
RL
8-
I i. L-
321 -
1
I
-c
I
I
0,310 0,320 0,330 0,UO 0,350 0,360
~ c lmol/dm3 ,
Figure 13. Plot of line tension K of Newton black films and
sodium dodecyl sulfate solutions vs NaCl concentration as determined by the “criticalbubble method”. With this expression, slightly larger absolute values for K were calculatedM(Figure 13) for the same NaDoS solutions as those used in ref 44, 45, and 50. 5. Heterogeneous Phase Formation Equation 3 as well as the theory of condensation from supersaturated vapors upon a substrate (heterogeneous phase formation), considered in section 3, indicates that this process will be particularly sensitive to the presence of the line tension K because of the extremely small size of nuclei of the new phase. Especially, if K is negative, reduction of the critical supersaturation and a change of the phase formation mechanism to barrierless condensation (W= 0) ought to be expected. Verification of these predictions was carried out by the method of the reverse Wilson chamber (RWC). The successive improvements of the RWC method of measuring the critical s ~ p e r s a t u r a t i o nfor ~ ~ water condensing on hexadecane brought the device and the measurements to a highly satisfactory state described in ref 59. Concurrently, the success of the investigation was enhanced69by the use of very pure hexadecane as the substrate and of extremely pure argon as the inert gas saturated with water vapor. These investigations are actually a further development of the work of Mitchel,w)who critically discussed the results of previous measurements for water condensation on a substratee1and who first introduced a liquid substrate. The latter is of basic importance, since the presence of nonhomogeneities (more active, better wettable regions) on a solid substrate could also cause a reduction of the critical supersaturation though in a less defined and more irreproducible manner. For example, this is Tohmfor and Volmer’s2eexplanation of the reduced critical overvoltage (supersaturation) obtained by Erdey-GrBz and Wicks2 for the electrolytic deposition of mercury on a (58) Platikanov, D.;Nedyalkov, M. Microscopic Aspects of Adhesion and Lubrication; Georges, J. M., Ed.; Elsevier: Amsterdam, 1982; p 97. (59) Chakarov, V.;Zembala, M.; Novozhilova, A.; Scheludko, A. Colloid Polym. Sci. 1987, 265, 347. (60) Mitchel, D. F. Ph.D. Thesis, Clarkeson College of Technology, 1976. (61) Koutaky, J. A.; Walton, A. G.; Baer, E. Surf. Sci. 1965, 3, 165. Jsaka,C. Ph.D. Thesis, University of Clermont-Ferrand, 1972. Twomey, J. J. Chem. Phys. 1959,30,941. Mahata, P. C.; Alofs, D. J. J. Atmos. Sci. 1976, 32, 116. (62) Erdey-GrCiz, T.; Wick, H. 2.Phys. Chem. 1932, A162,62.
Figure 14. Diagram of the reverse Wilson chamber (RWC) for condensation on a substrate with photoelectric detection of the condensate.
polished graphite electrode. An analogous exanation was put forward by Scheludko and Todorovae3 for the electrolytic deposition of mercury on a polycrystalline platinum microcathode. In ref 28 the more precise data of ref 63 on the lowering of the critical supersaturation are reinterpreted as an effect of a negative line tension K , requiring K = -10” dyn. The RWC is shown schematically in Figure 14. Due to compression of the water-saturated inert gas in chamber a with piston b, the partial pressure of water is increased above that of the saturated vapor; i.e., supersaturation is created with respect to the chamber wall and the hexadecane covering window d. The latter is observed through the microscope M, and the light is registered with the photodetector EMI. In the absence of condensation, the observed field is almost completely dark, due to phototrap f, while at the onset of condensation, photocurrent pulses are detected in the EM1 from the light scattered by the droplets formed upon the hexadecane surface. These impulses increase linearly upon increasing the degree of compression and, therefore, the supersaturation. A detailed description of the latest version of the setup is given in ref 59. The exact conditions of the optimum regime of compression (intermediate between adiabatic and isothermal) as well as the manner of calculating the supersaturation are presented there. The critical supersaturation value is determined by extrapolating to zero the photopulses. This point corresponds to the onset of the intensive barrierless condensation and, hence, marks the transition from barrier-controlled (unnoticeable) to barrierless formation of droplets. In ref 59 and the last paper of ref 31, the critical supersaturation is interpreted in terms of the effect of K , for the more general and more complex case of a doubleconvex droplet on a liquid substrate. However, the calculations are, in principle, not different from those for the simpler case of a plane (solid) substrate, as presented in section 3 and, therefore, will not be discussed here. In Figure 15, the latest version of RWC is shown. Compression is preset with air under known pressure by opening the valve e, and the total time of compression is accurately measured with electronic clock g. In the thermostated buffer vessel 12, the inert gas is saturated in advance with the vapor of a KI solution of known concentration. The temperature of the substrate is equalized to within 0.01 “C with that of the KI solution 15 at the botton of 14 and is controlled by thermocouple ~~
(63) Scheludko, A.;Todorova, M. Zzv. Bulg. Akad. Nauk., Otd. Fir.Mat. Tech. Nauki, Ser. Fir. 1962, 3,61.
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498 Langmuir, Vol. 4, No. 3, 1988 1’5 v
voltmeter or r-t recorder
Figure 15. Diagram of the RWC system for measuring the critical supersaturation (for the details see ref 59).
4. After this, the gas compression takes place and the recording of the photopulses is also started. The device is separately calibrated with respect to the P / V regime by replacing the window with an elastic membrane, the deformation of which is registered with the aid of a sensitive transducer. Thus the pressure increase corresponding to the preset time of gas compression is measured. Measurements conducted with very pure argon showed that the critical supersaturation is completely independent of the degree of initial undersaturation determined by the concentration of the KI solution. The value of the critical = 0.204 supersaturation thus obtained was In instead of the value of 0.755 expected from the theory, which disregards K. Finally, in ref 59, the effect of warming of the substrate at the regime of compression employed (intermediate between adiabatic and isothermal) was estimated; it turns out to be negligible, which makes superfluous the corresponding correction of the supersaturation. In this way the exact determination of K for water condensing on hexadecane from saturated argon was made possible. A value of K = -1.9 X dyn was obtained for the onset of barrierless condensation. This final result is in qualitative agreement with the previous approximate data obtained in ref 31 and lends support to the view that a negative K is responsible for the substantial reduction of the critical supersaturation, that the barrier-controlled mechanism of condensation in the bulk of the supersaturated vapors is replaced by the barrierless mechanism for the condensation on a substrate, and that the value of K is of the order of magnitude of -10” dyn, i.e., is very low. 6. Conclusion The line tension problem has undergone a variety of developments in both theoretical and experimental directions. “Linear” thermodynamics, although not developed to the completeness of “surface” thermodynamics, has reached categoric and clearly stated conclusions with respect to the mechanical equilibrium of the three-phase contact and the energy barrier in the formation of nuclei
of a new phase on a substrate. Less clear are the results of model calculations of line tension, mainly because of the poor knowledge of shortrange interaction effects in thin films.Thus the calcualtion of K for the perimeter of a thicker film forming the contact zone with a non-zero contact angle, where K is extremely low, turns out to be relatively reliable, while for a very thin Newton black film and a bare surface, where K is considerably higher, such an estimate is more uncertain. In any case, all these estimates yield very low K values, less than lo+ dyn and usually negative, although the reversal of its sign is not excluded. These difficulties in the theoretical estimation of K make its experimental determination even more important and significant. The difficulties are considerable because of the low K values. The methods employed for this task are either direct, based on formula 3 and a determination of the dependence of the contact angle on the radius of the three-phase contact perimeter (when this perimeter is of microscopic size), or indirect, the value of K is obtained from the critical size of a bubble (or a solid sphere) or from the critical supersaturation for the condensation on a substrate. With the direct methods the effect of K (or of K/r) is much weaker and can easily be masked, partially or completely, by nonequilibrium (hysteresis) phenomena. This is why a number of previous attempts to study the variation of the contact angle with the perimeter of the threephase contact, especially at larger r, have led to doubtfully high K values, often called pseudoline tension. Since, according to formula 3, K/r is usually a small difference of the projections of the q,these K/r values must be known very accurately in order to obtain reliable values. Otherwise, uncertain and even false results can be obtained. The most reliable data for K obtained by this approach come from the study of Newton black films of NaDoS solutions, where there are no indications of any hysteresis of the contact angle and the surface tensions of the solution and of the Newton f i i are accurately known from various independent sources. From these measurements, the dependence of K on NaCl concentration is obtained, with K
Langmuir 1988,4,499-511 changing sign and varying from +lo4 (through zero) to -lo4 dyn. The positive branch of this dependence is separately c o n f i i e d by the indirect method of the critical bubble of microscopic r. This suggests that the values of K are reliable. The most sensitive indirect method used, namely, that of measuring the critical supersaturation for the condensation of water vapor on hexadecane, has led to a reliable determination of K in this system: K = -1.9 X lob dyn. The only uncertain point in this case is that, with nuclei (droplets) of molecular size, some effect of surface curva-
499
ture on surface tensions (or, for that matter, the effect of the curvature of the three-phase contact line on K ) cannot be excluded. As is known, this difficulty is typical for all cases of homogeneous or heterogeneous phase formation. The state of the line tension problem, thus outlined, shows that it emerges as a promising and important, although arduous, part of the physical chemistry of dispersed systems.
Acknowledgment. The assistance of K. J. Mysels in editing this paper is acknowledged.
General Patterns of the Phase Behavior of Mixtures of H20, Nonpolar Solvents, Amphiphiles, and Electrolytes. 1 M. Kahlweit,* R. Strey, P. Firman, D. Haase, J. Jen, and R. Schomacker Max-Planck-Institut fur Biophysikalische Chemie, Postfach 2841,D-3400Gbttingen, FRG Received November 20, 1987 The phase behavior of mixtures of water, oils, nonionic amphiphiles, and electrolytes (microemulsions) follows general patterns that originate from the interplay between the lower miscibility gap of the binary mixture oil-amphiphile and the upper (closed) miscibility gap of the binary mixture water-amphiphile (section II). As a consequence,the critical line, connecting the plait pointa of the isothermal phase diagrams of the ternary mixtures, changes from the ail-rich to the water-rich side with rising temperature. If the chemical potentials of the components are appropriately changed, the critical line may be “broken” at a tricritical point, which gives rise to the evolution of a three-phase body (111). In the amphiphile-rich phase of the three-phase body (the microemulsion),one finds-for thermodynamic reasons-a maximum of the mutual solubility between water and oil combined with a minimum of the interfacial tension between the aqueous and the oil-rich phase (VII). Since the position and extensionsof the three-phase bodies are strongly correlated with those of the binary miscibility gaps, they show the same systematic dependence on the nature of the oils and the amphiphiles (VI), on pressure (VIII), and on the concentration of an added electrolyte (1x1. In this review we summarize the general patterns of this dependence, the knowledge of which is indispensable for applying such mixtures in research and industry.
I. Introduction Mixtures of water, nonpolar solvents (oils), amphiphiles, and inorganic electrolytes may separate into three liquid phases, an aqueous (a), an amphiphile-rich (c), and an oil-rich phase (b). In phase c, often referred to as “microemulsion”, one finds-for thermodynamic reasons-a maximum of the mutual solubility between water and oil. The three-phase body exists only within a well-defined temperature interval. Near the mean temperature of that interval one finds-again for thermodynamic reasons-a minimum of the interfacial tension between the aqueous and the oil-rich phase. The position of the three-phase body on the temperature scale as well as ita extensions depends sensitively on the chemical nature of the components and the concentration of the electrolyte. The detailed knowledge of this dependence is a prerequisite not only for performing meaningful experiments in order to clarify the properties of such mixtures but also for applying them in research and industry. In industry, in general, the temperature, the oil, and the composition of the aqueous solution (brine) are given. Wanted is the most efficient amphiphile, i.e., that amphiphile the addition of which leads to the highest mutual solubility between that particular oil and that particular brine at that particular temperature. That is why studying the dependence of the properties of the three-phase body on the chemical nature of the components of the mixture is so important. Experience has shown that the phase
behavior of such mixtures follows general patterns. The knowledge of these patterns permits predicting the mean temperature of the three phase body as well as the efficiency of the amphiphile and thus which amphiphile (or which combination of amphiphiles) to apply for solving a particular problem. In this paper we shall, therefore, summarize an empirical (!) description of these patterns. For reasons that will become clear later it is appropriate to distinguish between nonionic and ionic amphiphiles. Accordingly, we shall in part 1 of this series consider quaternary mixtures H20 (A)-“oil” (B)-nonionic amphiphile ((2)-inorganic electrolyte (E), first considering ternary mixtures A-B-C and then the quarternary mixtures. Mixtures including ionic amphiphiles will be presented in part 2.
11. Critical Line A mixture of m components has-in the absence of external fields-m + 1independent thermodynamic variables, namely, the temperature T, the external pressure p, and m - 1 composition variables, the choice of which is a matter of convenience. Since experience shows that the effect of pressure is weak compared with that of temperature, we shall, in general, dispense with p by keeping it constant at atmospheric pressure. The phase behavior of a ternary mixture may then be represented exactly in an upright phase prism with the Gibbs triangle A-B-C as the base and T as the ordinate (Figure 1). As for composition
Q743-7463/8a/2404-0499$01.50/0 1988 American Chemical Society
