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Penetration — a comparison of various models
Colloids and Surfaces A: Physicochemical and Engineering Aspects. 16 (1993) 197-201 Elsevier Science Publishers B.V., Amsterdam
Penetration -
197
a comparison of various models
S. Siegel and D. Vollhardt Max-Planck-lnstitutfiir
Kolloid- und Grenzjltichenforschung, Rudower Chaussee 5, 12 489 Berlin, Germany
(Received
1992; accepted
28 September
29 January
1993)
Abstract Equilibrium penetration has been studied for the sodium has been calculated in terms of the monolayer density penetration models. The results have been compared consider the area covered by the monolayer molecules. a wide range of validity, thus allowing good estimation Keywords: Models; penetration;
sodium dodecyl
dodecyl sulphateeoctadecanol system. The surface composition and the surfactant concentration on the basis of three different and discussed. The model with least approximation does not By introducing a geometrical factor the equation obtained has of the surface composition.
sulphate-octadecanol
Introduction The penetration of insoluble monolayers by water-soluble species is of increasing relevance to the study of a multitude of interfacial systems [l-3]. For example, information on biological membranes or on the interaction between two surfactants can be obtained from monolayer penetration studies. This is the reason for a number of papers in which the interactions of various lipids and proteins with surfactants have been studied experimentally. In fact, three important models are available for obtaining quantitative information on the molecular composition of the penetrated monolayer in equilibrium. However, in only a few papers were the models applied to experimental data, mostly for particular conditions. Nevertheless, a number of open questions remain regarding the understanding and modelling of adsorption on a surface covered by a monolayer. The resulting surface tension, which is lower than the sum of the surface pressure lowering effects of the two Correspondence to: S. Siegel, Kolloidund Grenzflachenforschung, 12 489 Berlin, Germany. 0927-7757/93/$06.00
0
1993 -
Max-Planck-Institut fur Rudower Chaussee 5,
Elsevier Science Publishers
system.
components, can barely be modelled, and the composition of the penetrated monolayer remains unknown. In this work, results of penetration measurements on a particular system with wide ranges of soluble surfactant concentration and monolayer coverage have been used to compare three theoretical models with different validity conditions. Experimental To obtain quantitative information about the penetrated monolayer under equilibrium, attention has to be focused on the experimental technique. In particular, incomplete mixing of the subphase in the trough after injecting the surfactant may be a reason for poor reproducibility [4]. A circular PTFE trough was equipped with a Wilhelmy-type surface balance (glass plate), and temperature monitoring and nitrogen rinsing systems. The principle of the method used consists of moving an insoluble monolayer enclosed between two barriers over different segments of the multicompartment trough. At first the monolayer is spread and compressed on a water subphase, B.V. All rights reserved.
5’. Siegel, D. Vollhardt/Colloids
198
Surfaces A: Physicochem.
Eng. Aspects
76 (1993)
197-201
and it is then swept onto the segment containing the soluble component [S]. Injection and stirring are not necessary; the subsolution is homogeneous
surface pressure 7~is the total reduction of the pure water surface tension due to both the soluble surfactant and the insoluble monolayer. From the
from the very beginning. details are given elsewhere
measured data a set of equivalent surface pressureconcentration isotherms, dependent on the monoand are layer coverage A,, can be constructed
The the
experimental
system
data
sodium
Further [6]. in this
dodecyl
experimental work
sulphate
refer
to
(SDS))
given in Fig. 2.
octadecanol at 22°C. The monolayer is spread on doubly distilled water from a lop3 M heptane solution, compressed, and manipulated to the
Models
segment expanding
Pethica penetration
containing the SDS solution. After the monolayer to a maximum area, it
is compressed again and the surface pressure-area (n--A) isotherm is recorded. Because the desorption of the surfactant and the relaxation of the monolayer are relatively fast, a compression rate of per molecule gives the same iso0.04 nm’ min- ’ therm as a stepwise compression. Therefore all isotherms are measured in a compression run with the given rate; the adsorption of the surfactant on the monolayer covered surface is very slow, and an expansion cycle would take a long time. The result of the above measurements is a set of z-A isotherms for subsolutions with different SDS concentrations; some are shown in Fig. 1. The
was the first to describe equilibrium by means of thermodynamics in 1955
[7]. He applied the Gibbs adsorption equation penetration experiments, thus obtaining AM
where A, is the total area per monolayer molecule, & the partial molecular area, c, the surfactant concentration, r, the surface excess concentration, R the gas constant and T the temperature. The partial molecular areas of monolayer and surfactant molecules are defined by A = n,A,
= nM& + II,&
40
40
II
1
I” 5
2
9): 4
30
30 E
E 2
2 E
E zo d ._ F
20
d .d E
10
10
0.2
area
per
10-e
0.3
molecule
in
nma
Fig. 1. Pressure-area isotherms of octadecanol at 22°C on water (left-hand curve) and on SDS solutions: curve 1, 2, 4. 10m4 mol 1-l; curve 3, 2. 1O-4 mot 1-r; curve and curve 5, 7.10-“mall-‘; curve 4, l.l~10~3moll~1; 1.4. 10m3 mall-‘.
I
,
0
0 0.1
to
10-4
concentration
10-s
in
10-Z
mol/l
Fig. 2. Pressure-concentration isotherms of SDS without monolayer (right-hand curve) and with monolayer coverages A, (area per monolayer molecule): curve 1, 0.26 nm*; curve 2, 0.22 nm*; curve 3, 0.20 nm2; curve 4, 0.19 nm’; curve 5, 0.184 nm’.
S. Siegel, D. Vollhardt/Colloids Surfaces A: Physicochem. Eng. Aspects 76 (1993)
199
197-201
where A is the total area and n is the number of molecules within the Gibbs surface, subscripts M and s indicating monolayer and surfactant, respectively. The number nM is constant and known. In a more general treatment based on Gibbs equation, Alexander and Barnes derived [S] 67L
( > 6 In c, A
RTA,s (AM - &,).&,(A,
- ;r, + 1,)
1-I(2)
0.0
concentration
This equation is based on the assumption of an ideal dilute solution of the surfactant in the monolayer as well as a constant Ahl at constant 71. Proceeding from this approximation, the penemonolayer at tration of an octadecanol z = 30 mN m-l by sodium dodecyl sulphate was analysed theoretically and the penetration isotherm was calculated [6]. In 1982, Motomura et al. [9] derived an integral equation (Eqn (3)) on the basis of his thermodynamics of mixed monolayers [lo] and the surface excess quantities defined by Hansen [l 11. This equation does not need any assumptions and should be valid over a wide range:
in
mol/l
Fig. 3. Calculated surface excess concentrations, r,, of SDS in the presence of an octadecanol monolayer, according to Eqn (1); the parameter is A, (nm2 per molecule).
2.0 -
0.0
m
s
10-D
@An)
SInAM
(3)
AM
where An = n - IX,- nM and I’,(a) designates the equilibrium surface excess concentration in the absence of the monolayer (AM+ 00). TerMinassian-Saraga derived a similar expression by another method in 1985 [12]. The results of the three theories applied to the experimental data of the system sodium dodecyl sulphate-octadecanol are shown in Figs 3-5. The parameter is the area (nm2) per monolayer molecule (AM).
lo-'
10-a
concentration Fig. 4. As Fig. 3, according
in
mol/l
to Eqn (2).
Results and discussion
Pethica’s equation corresponds to a simple geometrical model with a narrow range of validity; it is only valid in the case of saturation adsorption (for SDS c, > 3 - 10e4 mol 1-l). At lower concentrations the calculated I-, values were too high (see Fig. 3). However, Eqn (1) is a simple approximation that is often used to estimate the surface composition of penetrated systems.
S. Siegel, D. Vollhardt/Colloids
200
Surfaces
A: Physicochem.
Eng. Aspects 76 (1993)
the monolayer (and not only beneath estimated from
Equation
(3) only considers
197-201
it), r, can be
the interaction
between
the surfactants and does not include the area covered by the monolayer molecules. By introducing a “geometrical
factor” according
to Eqns (4) or
(1) into Eqn (3) it follows that AM-AM A M
r,=r,(c0) concentration Fig. 5. As Fig. 3, according
in
--
1 RTAM s
Equation (2) should be valid at high monolayer coverage [S], especially for a highly incompressible But for such as monolayer octadecanol. A, < 0.21 nm* (the region where rcM> 0) and saturation adsorption, the calculated I-, values in our system are unreasonable, because they have a maximum. Figure 4 shows the calculated & values for higher AH values. In Eqn (3) the assumption of constant A, at constant pressure in pure and mixed systems has not been used. At lower monolayer coverage and high cs, r, z rs(co), whereas at lower concentration a reduced surface concentration of SDS dependent on A, is expected. At lower monolayer coverage and higher c, values the calculated r, z r,(a) values are reasonable; however, the calculated r, values do not depend on A, but only on the effect of the monolayer on the slope &-c/~ In c, of the isotherm. This means that r, is smaller than T,(co) only when the slope of the isotherm is influenced by the monolayer. The experiment has proved that this is the case for AM < 0.3 nm* (Fig. 2). The crosssection 2, of an octadecanol molecule is about 0.18 nm* and therefore the monolayer coverage at A, = 0.3 nm* is already about 60%. But from Eqn (3) it still follows that r, z r,(co). Hence the geometrical area has to be reduced to the free area by a correction factor. If there is penetration into
dA,
(5)
AM
mol/l
to Eqn (3).
E
Unlike Eqn (3) this equation leads to results which are closer to those expected at lower c, and lower A, values. The influence of both c, and A, is considered in the new Eqn (5) (Fig. 5). Figure 6 shows the calculated values for r, according to the different models. For comparison, the surface excess concentration of SDS in the pure system without a monolayer calculated with the Gibbs equation is also given. As this value has to be the upper limit for reasonable r, values in the penetrated system, it is obvious that Eqn (1) is only valid
AM=
for higher
0.27
values
of c,. Equations
(2)
mn2
10-d
concentration Fig. 6. Comparison of the calculated SDS at A, = 0.27 nm’ per molecule. equations used.
10-S
in
10-Z
mol/l
surface concentrations The numbers indicate
of the
S. Siegel, D. Vollhardt/Colloids Surfaces A: Physicochem. Eng. Aspects 76 (1993)
and (5) yield comparable values; the results of Eqn (3) are very close to those of adsorption on a free surface. In Fig. 7, for further clarity, curve 2 is not shown. Here, at a higher monolayer coverage of Ahl = 0.22 nm* per molecule, Eqns (l), (2) and (5) give comparable results, whereas the results of Eqn (3) are again higher than the upper limit estimated by Eqn (4). Figures 6 and 7 show convincingly that regarding the cross-section & of the monolayer molecule, the Eqn (5) introduced is valid at low monolayer coverage and for low surfactant concentrations as well as at higher coverage and for higher surfactant concentrations.
5
I
I
197-201
201
Conclusions
The three models given in the literature (Eqns (l)-(3)) are different with respect to range and conditions of validity. However, even within their validity ranges, the models give different values for the composition of the surface. The calculated r, values differ by a factor of l-5. The limiting cases A, + co, A, -+&, and c,+CMC, especially, are not included in the models. On introducing a geometrical factor into Eqn (3), better agreement between the models (2) and (3) is obtained, and an equation (Eqn (5)) valid over a wide range of c, and A, values, is obtained. With use of Eqn (5) a reasonable estimation of the surface excess concentration r, in penetrated systems can be made. References 1 2 3 4 5 6 7 8
1o-3
10-4
concentration
in
mol/l
Fig. 7. Comparison of the calculated concentrations of SDS, at AM = 0.22 nmz per molecule. The numbers indicate the equations used.
9 10 11 12
J.H. Schulman and A.H. Hughes, Biochem. J., 29 (1935) 1243. G. Colacicco, Lipids, 5 (1970) 636. Y. Hendrikx and L. Ter-Minassian-Saraga, Adv. Chem. Ser., 144 (1975) 177. M.A. McGregor and G.T. Barnes, J. Colloid Interface Sci., 60 (1977) 408. D. Vollhardt, Mater. Sci. Forum, 25, 26 (1988) 541. D. Vollhardt and M. Wittig, Colloids Surfaces, 47 (1990) 233. B.A. Pethica, Trans. Faraday Sot., 51 (1955) 1402. D.M. Alexander and G.T. Barnes, J. Chem. Sot. Faraday Trans. 1, 76 (1980) 118. K. Motomura, Y. Hayami, M. Aratono and R. Matuura, J. Colloid Interface Sci., 87 (1982) 333. K. Motomura, J. Colloid Interface Sci., 48 (1974) 307. R.S. Hansen, J. Phys. Chem., 66 (1962) 410. L. Ter-Minassian-Saraga, Langmuir, 1 (1985) 391.
Penetration -
197
a comparison of various models
S. Siegel and D. Vollhardt Max-Planck-lnstitutfiir
Kolloid- und Grenzjltichenforschung, Rudower Chaussee 5, 12 489 Berlin, Germany
(Received
1992; accepted
28 September
29 January
1993)
Abstract Equilibrium penetration has been studied for the sodium has been calculated in terms of the monolayer density penetration models. The results have been compared consider the area covered by the monolayer molecules. a wide range of validity, thus allowing good estimation Keywords: Models; penetration;
sodium dodecyl
dodecyl sulphateeoctadecanol system. The surface composition and the surfactant concentration on the basis of three different and discussed. The model with least approximation does not By introducing a geometrical factor the equation obtained has of the surface composition.
sulphate-octadecanol
Introduction The penetration of insoluble monolayers by water-soluble species is of increasing relevance to the study of a multitude of interfacial systems [l-3]. For example, information on biological membranes or on the interaction between two surfactants can be obtained from monolayer penetration studies. This is the reason for a number of papers in which the interactions of various lipids and proteins with surfactants have been studied experimentally. In fact, three important models are available for obtaining quantitative information on the molecular composition of the penetrated monolayer in equilibrium. However, in only a few papers were the models applied to experimental data, mostly for particular conditions. Nevertheless, a number of open questions remain regarding the understanding and modelling of adsorption on a surface covered by a monolayer. The resulting surface tension, which is lower than the sum of the surface pressure lowering effects of the two Correspondence to: S. Siegel, Kolloidund Grenzflachenforschung, 12 489 Berlin, Germany. 0927-7757/93/$06.00
0
1993 -
Max-Planck-Institut fur Rudower Chaussee 5,
Elsevier Science Publishers
system.
components, can barely be modelled, and the composition of the penetrated monolayer remains unknown. In this work, results of penetration measurements on a particular system with wide ranges of soluble surfactant concentration and monolayer coverage have been used to compare three theoretical models with different validity conditions. Experimental To obtain quantitative information about the penetrated monolayer under equilibrium, attention has to be focused on the experimental technique. In particular, incomplete mixing of the subphase in the trough after injecting the surfactant may be a reason for poor reproducibility [4]. A circular PTFE trough was equipped with a Wilhelmy-type surface balance (glass plate), and temperature monitoring and nitrogen rinsing systems. The principle of the method used consists of moving an insoluble monolayer enclosed between two barriers over different segments of the multicompartment trough. At first the monolayer is spread and compressed on a water subphase, B.V. All rights reserved.
5’. Siegel, D. Vollhardt/Colloids
198
Surfaces A: Physicochem.
Eng. Aspects
76 (1993)
197-201
and it is then swept onto the segment containing the soluble component [S]. Injection and stirring are not necessary; the subsolution is homogeneous
surface pressure 7~is the total reduction of the pure water surface tension due to both the soluble surfactant and the insoluble monolayer. From the
from the very beginning. details are given elsewhere
measured data a set of equivalent surface pressureconcentration isotherms, dependent on the monoand are layer coverage A,, can be constructed
The the
experimental
system
data
sodium
Further [6]. in this
dodecyl
experimental work
sulphate
refer
to
(SDS))
given in Fig. 2.
octadecanol at 22°C. The monolayer is spread on doubly distilled water from a lop3 M heptane solution, compressed, and manipulated to the
Models
segment expanding
Pethica penetration
containing the SDS solution. After the monolayer to a maximum area, it
is compressed again and the surface pressure-area (n--A) isotherm is recorded. Because the desorption of the surfactant and the relaxation of the monolayer are relatively fast, a compression rate of per molecule gives the same iso0.04 nm’ min- ’ therm as a stepwise compression. Therefore all isotherms are measured in a compression run with the given rate; the adsorption of the surfactant on the monolayer covered surface is very slow, and an expansion cycle would take a long time. The result of the above measurements is a set of z-A isotherms for subsolutions with different SDS concentrations; some are shown in Fig. 1. The
was the first to describe equilibrium by means of thermodynamics in 1955
[7]. He applied the Gibbs adsorption equation penetration experiments, thus obtaining AM
where A, is the total area per monolayer molecule, & the partial molecular area, c, the surfactant concentration, r, the surface excess concentration, R the gas constant and T the temperature. The partial molecular areas of monolayer and surfactant molecules are defined by A = n,A,
= nM& + II,&
40
40
II
1
I” 5
2
9): 4
30
30 E
E 2
2 E
E zo d ._ F
20
d .d E
10
10
0.2
area
per
10-e
0.3
molecule
in
nma
Fig. 1. Pressure-area isotherms of octadecanol at 22°C on water (left-hand curve) and on SDS solutions: curve 1, 2, 4. 10m4 mol 1-l; curve 3, 2. 1O-4 mot 1-r; curve and curve 5, 7.10-“mall-‘; curve 4, l.l~10~3moll~1; 1.4. 10m3 mall-‘.
I
,
0
0 0.1
to
10-4
concentration
10-s
in
10-Z
mol/l
Fig. 2. Pressure-concentration isotherms of SDS without monolayer (right-hand curve) and with monolayer coverages A, (area per monolayer molecule): curve 1, 0.26 nm*; curve 2, 0.22 nm*; curve 3, 0.20 nm2; curve 4, 0.19 nm’; curve 5, 0.184 nm’.
S. Siegel, D. Vollhardt/Colloids Surfaces A: Physicochem. Eng. Aspects 76 (1993)
199
197-201
where A is the total area and n is the number of molecules within the Gibbs surface, subscripts M and s indicating monolayer and surfactant, respectively. The number nM is constant and known. In a more general treatment based on Gibbs equation, Alexander and Barnes derived [S] 67L
( > 6 In c, A
RTA,s (AM - &,).&,(A,
- ;r, + 1,)
1-I(2)
0.0
concentration
This equation is based on the assumption of an ideal dilute solution of the surfactant in the monolayer as well as a constant Ahl at constant 71. Proceeding from this approximation, the penemonolayer at tration of an octadecanol z = 30 mN m-l by sodium dodecyl sulphate was analysed theoretically and the penetration isotherm was calculated [6]. In 1982, Motomura et al. [9] derived an integral equation (Eqn (3)) on the basis of his thermodynamics of mixed monolayers [lo] and the surface excess quantities defined by Hansen [l 11. This equation does not need any assumptions and should be valid over a wide range:
in
mol/l
Fig. 3. Calculated surface excess concentrations, r,, of SDS in the presence of an octadecanol monolayer, according to Eqn (1); the parameter is A, (nm2 per molecule).
2.0 -
0.0
m
s
10-D
@An)
SInAM
(3)
AM
where An = n - IX,- nM and I’,(a) designates the equilibrium surface excess concentration in the absence of the monolayer (AM+ 00). TerMinassian-Saraga derived a similar expression by another method in 1985 [12]. The results of the three theories applied to the experimental data of the system sodium dodecyl sulphate-octadecanol are shown in Figs 3-5. The parameter is the area (nm2) per monolayer molecule (AM).
lo-'
10-a
concentration Fig. 4. As Fig. 3, according
in
mol/l
to Eqn (2).
Results and discussion
Pethica’s equation corresponds to a simple geometrical model with a narrow range of validity; it is only valid in the case of saturation adsorption (for SDS c, > 3 - 10e4 mol 1-l). At lower concentrations the calculated I-, values were too high (see Fig. 3). However, Eqn (1) is a simple approximation that is often used to estimate the surface composition of penetrated systems.
S. Siegel, D. Vollhardt/Colloids
200
Surfaces
A: Physicochem.
Eng. Aspects 76 (1993)
the monolayer (and not only beneath estimated from
Equation
(3) only considers
197-201
it), r, can be
the interaction
between
the surfactants and does not include the area covered by the monolayer molecules. By introducing a “geometrical
factor” according
to Eqns (4) or
(1) into Eqn (3) it follows that AM-AM A M
r,=r,(c0) concentration Fig. 5. As Fig. 3, according
in
--
1 RTAM s
Equation (2) should be valid at high monolayer coverage [S], especially for a highly incompressible But for such as monolayer octadecanol. A, < 0.21 nm* (the region where rcM> 0) and saturation adsorption, the calculated I-, values in our system are unreasonable, because they have a maximum. Figure 4 shows the calculated & values for higher AH values. In Eqn (3) the assumption of constant A, at constant pressure in pure and mixed systems has not been used. At lower monolayer coverage and high cs, r, z rs(co), whereas at lower concentration a reduced surface concentration of SDS dependent on A, is expected. At lower monolayer coverage and higher c, values the calculated r, z r,(a) values are reasonable; however, the calculated r, values do not depend on A, but only on the effect of the monolayer on the slope &-c/~ In c, of the isotherm. This means that r, is smaller than T,(co) only when the slope of the isotherm is influenced by the monolayer. The experiment has proved that this is the case for AM < 0.3 nm* (Fig. 2). The crosssection 2, of an octadecanol molecule is about 0.18 nm* and therefore the monolayer coverage at A, = 0.3 nm* is already about 60%. But from Eqn (3) it still follows that r, z r,(co). Hence the geometrical area has to be reduced to the free area by a correction factor. If there is penetration into
dA,
(5)
AM
mol/l
to Eqn (3).
E
Unlike Eqn (3) this equation leads to results which are closer to those expected at lower c, and lower A, values. The influence of both c, and A, is considered in the new Eqn (5) (Fig. 5). Figure 6 shows the calculated values for r, according to the different models. For comparison, the surface excess concentration of SDS in the pure system without a monolayer calculated with the Gibbs equation is also given. As this value has to be the upper limit for reasonable r, values in the penetrated system, it is obvious that Eqn (1) is only valid
AM=
for higher
0.27
values
of c,. Equations
(2)
mn2
10-d
concentration Fig. 6. Comparison of the calculated SDS at A, = 0.27 nm’ per molecule. equations used.
10-S
in
10-Z
mol/l
surface concentrations The numbers indicate
of the
S. Siegel, D. Vollhardt/Colloids Surfaces A: Physicochem. Eng. Aspects 76 (1993)
and (5) yield comparable values; the results of Eqn (3) are very close to those of adsorption on a free surface. In Fig. 7, for further clarity, curve 2 is not shown. Here, at a higher monolayer coverage of Ahl = 0.22 nm* per molecule, Eqns (l), (2) and (5) give comparable results, whereas the results of Eqn (3) are again higher than the upper limit estimated by Eqn (4). Figures 6 and 7 show convincingly that regarding the cross-section & of the monolayer molecule, the Eqn (5) introduced is valid at low monolayer coverage and for low surfactant concentrations as well as at higher coverage and for higher surfactant concentrations.
5
I
I
197-201
201
Conclusions
The three models given in the literature (Eqns (l)-(3)) are different with respect to range and conditions of validity. However, even within their validity ranges, the models give different values for the composition of the surface. The calculated r, values differ by a factor of l-5. The limiting cases A, + co, A, -+&, and c,+CMC, especially, are not included in the models. On introducing a geometrical factor into Eqn (3), better agreement between the models (2) and (3) is obtained, and an equation (Eqn (5)) valid over a wide range of c, and A, values, is obtained. With use of Eqn (5) a reasonable estimation of the surface excess concentration r, in penetrated systems can be made. References 1 2 3 4 5 6 7 8
1o-3
10-4
concentration
in
mol/l
Fig. 7. Comparison of the calculated concentrations of SDS, at AM = 0.22 nmz per molecule. The numbers indicate the equations used.
9 10 11 12
J.H. Schulman and A.H. Hughes, Biochem. J., 29 (1935) 1243. G. Colacicco, Lipids, 5 (1970) 636. Y. Hendrikx and L. Ter-Minassian-Saraga, Adv. Chem. Ser., 144 (1975) 177. M.A. McGregor and G.T. Barnes, J. Colloid Interface Sci., 60 (1977) 408. D. Vollhardt, Mater. Sci. Forum, 25, 26 (1988) 541. D. Vollhardt and M. Wittig, Colloids Surfaces, 47 (1990) 233. B.A. Pethica, Trans. Faraday Sot., 51 (1955) 1402. D.M. Alexander and G.T. Barnes, J. Chem. Sot. Faraday Trans. 1, 76 (1980) 118. K. Motomura, Y. Hayami, M. Aratono and R. Matuura, J. Colloid Interface Sci., 87 (1982) 333. K. Motomura, J. Colloid Interface Sci., 48 (1974) 307. R.S. Hansen, J. Phys. Chem., 66 (1962) 410. L. Ter-Minassian-Saraga, Langmuir, 1 (1985) 391.