Langmuir 1987, 3, 725-729
Vc ‘Reference’ cmJlg
Figure 7. Comparison plots for adsorption (a) and desorption (b) isotherms for Aerosil 150. The data for the 1-10-mm agglomerates are plotted as filled points, and the 1-10-pm agglomerate data are plotted as open points. 0.6
i.4
0
0.08 0
0.08
Vc ‘Comparison’ cm3Ig
Figure 8. Comparison plots for adsorption (a) and desorption (b) isotherms for AerosilOX50. The data for the 1-10.” agglomerates are plotted as filled points, and the 1-10-pm agglomerate data are plotted as open points.
size of moderately dense aggregates of spherical particles without disturbing the particle packing. The largest particles (Aerosil OX50),having more than twice the spherical-equivalent diameter of the next largest particles, are apparently susceptible to either particle fracture,
725
particle displacement, or both. Packings of the smallest particle materials (Aerosil380) may also be altered by the size reduction process. Both mercury penetration pore size distributions and nitrogen comparison plots show evidence of significant intraagglomerate meso- and macroporosity in the small agglomerate materials. This is not surprising given the small size (between 1and 10 Hm) of these agglomerates. For nitrogen adsorption, this porosity introduces difficulties in quantitative analysis. For example, size distribution calculations are not strictly applicable to the isotherms measured on the small agglomerate materials, since no Gurvitsch plateau is exhibited. If these calculations are performed anyway, one finds that the desorption pore size distributions obtained for small-agglomeratematerials are remarkably similar to the corresponding adsorption pore size distributions, whereas for the large agglomerate materials, the desorption distributions were much sharper? In spite of the uncertainties associated with the additional intraagglomerate porosity, the presence of the network effect on desorption is strongly indicated. Analysis of the adsorption comparison plots suggests similar mesopore structures in both small and large agglomerate materials, while the desorption plots show the transition which would be expected if a network-type constraint was relaxed upon size reduction. It is much more difficult to construct an alternate explanation based upon changes in the interagglomerate pore structures to explain these trends. And the mercury penetration data, showing peak pore sizes which are identical for both agglomerate sizes, give a very positive indication that the interagglomerate mesopore structures are similar and that the major difference is due to a network effect in the large agglomerates.
Acknowledgment. This work was performed at the Kentucky Energy Cabinet Laboratory (KECL) with funding from the Commonwealth of Kentucky, Kentucky Energy Cabinet. KECL is operated under contract by the University of Louisville. Registry No. Aerosil, 7631-86-9; N, 7727-37-9.
Influence of Clustering on the Behavior of Insoluble Monolayers on Water: A Theoretical Approach J. A. Poulis, A. A. H. Boonman,? P. Gieles,t and C. H. Massen* Physics Department, Eindhoven University of Technology, 5600 MB Eindhouen, The Netherlands Received November 25, 1986. I n Final Form: February 11, 1987 A description is presented of surface pressure-area (*-A) curves of insoluble monolayers on water. It starts from the assumption that clusters of monolayer molecules exist, where the size of the clusters is governed by mutual equilibrium reactions. By introducing two parameters, the theoretical description deals with different phenomena encountered in r - A measurements without introducing phase transitions.
Introduction Comparison between pressure-volume (p-v) isotherms of gases and surface pressure-area ( r - A ) isotherms of insoluble monolayers has frequently influenced the models describing molecular and cooperative phenomena in m~nolayers.l-~Especially the van der Waals equation with Supported by the Netherlands Asthma Foundation. Supported by the Organization for the Advancement of Pure Scientific Research (Foundation for Biophysics). f
0743-7463/87/2403-0725$01.50/0
its ready introduction to phase transitions has often been referred to. In particular the F A curves show us, just like (1)Mingins, J.; Taylor, J. A. G.; Owens, N. F.; Brooks, J. H. Ado. Chem. Ser. 1975,144, 28. (2) Adam, N . K.; Jessom G. Proc. R.SOC.London, Ser. A 1926, 110, 423. (3) Dill, K. A.; Cantor, R. S. In Proceedings of the International Physics of Amphiphiles, Micelles, Vesicles and Microemulsions School of Physics “Enrico Fermi” (1983);Degiorgio, V., Corti, M., Eds.; NorthHolland Physics: Amsterdam, 1985; p 376.
0 1987 American Chemical Society
726 Langmuir, Vol. 3, No. 5, 1987
the van der Waals isotherms, smooth curves for small pressures. Also at larger compression both types of curves show anomalies in the slope which can range between hardly distinguishable and not differentiable. In the case of the a-A isotherms, these anomalies are frequently referred to as,e.g., phase transitions, squeeze-out effects, and collapse.4+ One of the differences between the surface pressure and gas pressure results is best shown by considering their slopes. In the a-A curves, the anomalies referred to as phase transitions, though showing unambiguously a maximum, do not usually attain zero or positive values, as is the case in the comparable p-V curve for a van der Waals gas. Albrecht et al.' brought the anomalies in the a-A curves in connection with the existance of clusters of monolayer molecules. In the present paper we shall outline how the *-A curves can be influenced greatly by clustering effects and so proffer a possible explanation for many of the anomalies. Thereto we shall follow the theory set up by Eisenberg and Tobolsky8v9who explained the polymerization in liquid sulfur and liquid selenium. Other work comparable to the polymerization theories is found in the field of micellization.1° Let us start from the van der Waals equation for a gas:
where p , V, n, and T stand for pressure, volume, number of molecules, and temperature, respectively; kB and NAv are the Boltzmann constant and Avogadro's number, respectively; b is the effective volume of a molecule; and a determines the pressure correction originating from the attractive forces between the gas molecules. Clustering of molecules will influence eq 1. When we restrict ourselves to the right-hand side (rhs) of eq 1,we may keep this expression unaltered in spite of clustering, when we adapt the meaning of the parameter n. Instead of the number of molecules we should use the number of clusters (including solitary molecules). Clustering like the (gas) pressure correction represented by the quantity a in eq 1 originates from attractive forces between the gas molecules. The two effects might therefore be interrelated. We shall ignore this coupling by restricting ourselves to the simple situaticn where only clustering occurs. That means that we shall use the equivalent of the van der Waals equation with a = 0 and get a(A - A,) = N,kBT (2) where A stands for the surface area, a for the surface pressure, Al for the sum of the minimum areas the molecules can cover, and N, for the number of clusters. In the following we shall, with the help of the polymerization theory, calculate how N , in eq 2 depends on A.
Poulis et al.
Nl equals the number of solitary molecules. Then we may write m
CiNi = No
(3)
i=l m
Ni = N,
(4)
i=l
where No stands for the total number of molecules available. As a reaction mechanism of the polymerization process we use (5) si + s1 F? Si+l where Sistands for a cluster consisting of i molecules. The reaction constant K, refers to the forward reaction, so
(%)-
=K-NiN,/A
When the reaction constant K, refers to the backward reaction, we obtain
(%)
c
A - A1 = - K N i + l y
The factor (A - Al)/A is caused by the requirement that for the backward reaction, area should be available to allow for the escape of solitary molecules. Using K = K,/K, we get for the equilibrium
Equation 8 leads to i-1
For the proposed calculation of N, it is useful to express N, and No as functions of N1 by using eq 9 into eq 3 and 4: (10)
and N, =
N1 KN1 1-A - A1
Equations 10 and 11hold under the condition 0 < K N l / ( A - A,) < 1,which reflects a monotonously decreasing value of Ni with increasing i; see eq 8. Solving N1 from eq 10 we get
Single-Parameter Treatment of Clustering In a simple description of clustering we shall introduce only one parameter, i.e., the equilibrium constant K. Let Nibe the number of clusters consisting of i molecules, so (4) Harkins, W. D. The Physical Chemistry of Surface Films; Reinhold: New York, 1954. (5) Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Interscience: New York, 1966. (6) Adamson, A. W. Physical Chemistry of Surfaces, 4th ed.; Wiley: New York, 1982. (7) Albrecht, 0.; Gruler, H.; Sackmann, E. J. Phys. 1978, 39, 301. (8) Tobolsky, A. V.; Eisenberg, A. J. J.Am. Chem. SOC.1959,81,780. (9) Poulis, J. A.; Massen, C. H.; Eisenberg, A.; Tobolsky, A. V. J.Am. Chem. Soc. 1965,87, 413. (10) Nagarajan, R. Adu. Colloid Interface Sei. 1986, 26, 205.
(7)
On combining eq 10 and 11 we obtain
From eq 12 and 13 we get
Langmuir, Vol. 3, No. 5, 1987 721
Influence of Clustering on Insoluble Monolayers
the specific molecular area A, of the molecules of the small clusters. Hereto we shall use the parameter G defined by A,, = A,(1 - G) (15) If we use Nomfor the number of molecules present in large clusters, we get
10
. Y
z
0.5
..."
Al = Ao( 1I
I
6
4
I
I
I
I
10
8
I
7 2.1 0-3
A
Figure 1. Influence of K on NJN0 when K = K , No = 10l6,and (-) K = 0; (--) K = 4 X lo-&;(---)K = 2 X A. = 4 X (...) K = 8 X 10-19; (-.-) K = 10-1'.
2.)
where A,, stands for the s u m of specific a r e a in the absence of large clusters. We also introduce the possibility that the equilibrium constant K has a different value Kk for the larger clusters. The calculation of N , becomes a little more complicated and it is useful to introduce the new variables Y and Yk defined by NIKk y = -N1K and Yk = (17) A - A1 A - A1 We get
Ni = YNi-, for i m Ni = YkNi-l for i 1 m Ni = N1Y-l for i < m
(18)
N,-l = N l P - 2
Ni = N l P - 2 Y i - m + 1for i 1 m With eq 18 and 19 we get
(19)
A
From eq 20 it follows that 0.061
N _O -
I
Nl
11 '\
n
1+ ( m - 1 ) P - m P - '
v2
I
8
I
(21)
Equations 21 and 22 hold under the condition 0 < Y < 1 and 0 < Yk < 1. To use eq 2 we still have to calculate N,. In analogy with eq 20 we get
I
10
A
Figure 2. (Top) Influence of K on T when Kk = K , No = 10l6, (-) K = 0; (--) K = 4 X (---) K = 2 and A = 4 X X K = 8 X 10-19; K = 1O-l'. (Bottom) Influence of K on ?r when Kk= 1.5K,No = 10l6,A. = 4 X loe3,G = 0, and (-) K = 4 X m = 20: (--) K = 0;(- - -) K = 2 X K = 8 x 10-19; (- .-) K = 10-17. (-a)
F - 2 Y k [ m- ( m - 1)Yk]
(1 - YkI2 (1 From the second term on the rhs of eq 21 we see
1 6
+
(-e-)
or
(-a)
Equation 14 allows us to calculate numerically Nc as a function of A. The results are shown in Figure 1. In Figure 2 (top) we see the a-A plot calculated from eq 2 with the data of Figure 1. Introduction of More Parameters Though there is a definite influence of the clustering mechanism on the *-A curve, the result in the preceding paragraph is not spectacular and does not allow for the explication of many anomalies of measured curves. To enlarge the possibilities of our model, we introduce a difference between large and small clusters. The molecules of the large clusters with number of molecules i > m will be given a specific molecular area A,, differing from
To get the resulting a-A curves we chose the following procedure. (i) Some parameters were given fixed values: kBT = 4 X J, No = A, = 4 X m2. (ii) The values of the other parameters are chosen differently for each of the a-A curves. These parameters were K , Kk, m, and G. (iii) For each a-A curve, a great number of values of Y were chosen arbitrarily and the corresponding values of A and a were calculated with the eq 16,17,21, 22, 24, and 2. In Figures 2-6 we see that by combining values of the different parameters we can get a great variety of a-A curves. It seems that a great part of the anomalies in the measured a-A curves can be covered by assuming clus-
Poulis et al.
728 Langmuir, Vol. 3, No. 5, 1987 I
I
I
1
I
I
I
1
i
4
-i 1
I
I
I
I
6
I
1
10.1oj
8
(-a)
Kk/K = 1.9;
(-e-)
I
Kk/K = 2.5.
I
6
i
Figure 3. Influence of K k / K on K when K = 4 X No = G = 0, and m = 2 0 (--) Kk/K = 1; ( - - - ) Kk/K A, = 4 X = 1.5;
I
I
A
A
Figure 6. Influence of G on ?r when K = 8 X K k / K = 1, No = 10l6,A, = 4 X W3,and m = 10: (-) G = 0; ( - - - ) G = 0.4.
vq
1
i ,
0.02
'\
t-
1
-..
A
1
-I
I
AW
Figure 7. K-A curve with the quantities used for the calculation of and eq8; see eq 27-29.
I
I
8.10''
6 A
Figure 4. Influence of m on ?r when K = 4 X lo-", Kk/K = 1.5, and G = 0 ( - - - ) m = 5; (-) m = 20; No = 10l6,A , = 4 X (-e)
m = 40.
t 1
I
I
!
I
assuming clustering. Here we refer to experiments where the area A is varied around a working point (Aw, a,) with a small amplitude and such high frequency that we may assume that Nc remains constant. This means that the cluster distribution remains constant. In this type of measurements it is customary to use the dilational elasticity E defined by E
= -A,Aa/AA
(25)
where Air and AA stand for the directly measured variations of a and A, respectively. Confining ourselves to situations where Air and AA are small enough to allow for the approximation AalAA = ds/dA, we obtain, with eq 2 E
O'
i
I
6 A
I
I
I
8.1V'
Figure 5. Influence of G on K when K = 4 X Kk/K = 1.5, and n = 20: (-) G = 0; G = 0.2.
No = 10l6,A. = 4 X
(-e)
tering. From Figures 3 and 4 we learn that plateaus become more pronounced with increasing Kk/K and m. For each of the many compounds and mixtures forming monolayers, the relevance of clustering should be discussed separately. So far, we only considered ir-A curves and we excluded hystereses and irreversibility effects one frequently encounters in experiments. We shall deal with some of these effects in the next paragraphs. Dilatational Elasticity So far we dealt with equilibrium situations. There is one more type of situation which can be dealt with easily by
= A,-
NckBT
A, - A.
A, --
A, - AOgw
(26)
When we restrict ourselves to very low frequencies (quasi-static) we can also calculate the value eqS of t from any x-A curve. We can use the same working point (Aw,ir,) but take for AhalAA not the directly measured value but the slope (dir/dA),, of the quasi-static s-A curve eqS = -Aw(
dir A, z ) B-AWaw qs
=
(27)
where the quantity B is found as depicted in Figure 7. From eq 26 and 27 we get
where the last expression refers to Figure 7. From the measured value of e, we can calculate A0 with eq 26.
Langmuir 1987, 3, 729-735
Assuming AI to be equal to A, gives us the possibility to calculate G from the combined t and T-A measurements.
Irreversible Processes The approach presented in the above could also contribute to the description of irreversible processes. Especially the squeeze-out effect could be treated well with the clustering mechanism. For squeeze-out effects, irreversible horizontal parts of the T-A curves are typical. Introducing clustering we could explain this behavior by assuming that there exists a modification of a certain cluster (“F”-cluster) containing F molecules which can disappear in the water below the surface layer. Thus we exclude squeeze-out in which molecules move to above the water-air interface. When the surface compression is performed slowly, once the clustering is gone so far that “F”-clusters appear by clustering, these clusters would transform into soluble clusters and disappear into the water causing a horizontal part in the T-A curve.
729
When the compression is going too fast, it might occur that not all the “F”-clusters are transformed into the soluble form and that bigger clusters are formed. This would lead to the return of the negative slope of the r A curve. A quantitative description of this phenomenon is the purpose of further work of our group.
Conclusions and Discussion We dealt with surface pressure-area (a-A) curves starting from the existence of clusters in the monolayer. Thereto we introduced the equilibrium clustering constant K. With this single parameter, however, we did not succeed in explaining plateaux in the (a-A) curves. The explanation was successful upon assuming that for larger clusters (exceeding m molecules) the equilibrium clustering constant differs from K. The resulting plateaux are more pronounced at high values of m. In addition a parameter G is introduced, accounting for the difference in specific area between molecules in large clusters and molecules in small clusters. With this theoretical description dif€erent phenomena encountered in 7r-A measurements can be coped with without introducing phase transitions.
Micellar Solubilization of 1-Pentanol in Binary Surfactant Solutions: A Regular Solution Approach C. Treiner,* A. Amar Khodja, and M. Fromon Laboratoire d%lectrochimie, U.A. 430 CNRS, Universite Pierre et Marie Curie, BAT.F., Paris 75005, France Received October 1, 1986. In Final Form: January 20, 1987 The partition coefficient P of 1-pentanol between micelles and water has been determined in the cases of a variety of mixed-surfactant solutions in the whole surfactant composition range by using gas chromatography. The following systems have been studied: (I) lithium dodecyl sulfate (LiDS) + lithium perfluorooctanesulfonate (LiFOS); (11) sodium dodecyl sulfate (SDS) + poly(oxyethy1ene) (23) dodecyl ether (POE23);(111) SDS + poly(oxyethy1ene) (4) dodecyl ether (POE4); (IV) sodium decyl sulfate (SDeS) + trimethyldecylammoniumbromide (Cl&r). The solubilizationof the solute in the mixed-surface solutions is calculated from the regular solution theory (applied to a three-component solution) by using the interaction parameter 6 deduced from the same theory as applied to the case of a binary surfactant solution. A positive departure from partition ideality is observed for system I; an almost ideal behavior is observed for system I1 and a larger departure from ideality for system 111; a strongly negative deviation from ideality is displayed with system IV. Thus the sign and magnitude of 0 govern the variation of P with micelle composition only in cases I and IV. In these binary surfactant solutions, 1-PeOH must be distributed throughout the micellar structure; however, in systems I1 and 111, the experimental results suggest a preferential solubilization in the mixed-micelle hydrocarbon core. An application of these findings to microemulsion formulations is suggested. Difficulties with the notion of micellar volume are outlined.
Introduction One of the main uses of surfactants concerns the solubilization in aqueous solutions of otherwise scarcely soluble compounds. A single surfactant can seldom practically achieve this purpose so that two or more surfactants are often needed. This situation is found most often in emulsion technology’ but is of more general concern. For emulsions, the HLB scale serves as a useful semiquantitative tool for the choice of a proper mixture of surfactants to emulsify a given oil; however, not much is known about the solubilization properties of mixed surfactants in homogeneous solutions. A few isolated examples may be
* To whom all correspondence
should be addressed.
0743-7463/87/2403-0729$01.50/0
found in the literature concerning micellaP4 or microemulsion solution^.^-^ A great deal of interesting theoretical work has been performed recently on the properties of non ideal mixed surfactant solutions.&l* These studies have prompted us (1) Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1985; Vol. 2. (2) Nishikido, N. J. Colloid Interface Sci. 1977, 60,242. (3) Tokiwa, F.;Tsujii, K. Bull. Chem. SOC.J p n . 1973, 46, 1338. (4) Treiner, C.;Vaution, C.; Miralles, E.; Puisieux, F. Colloids Surf. 1985, 14, 285. (5) Shincda, K.; Kuneida, H. J. Colloid Interface Sci. 1973, 42, 381. (6) Koukounis, C.;Wade, W. H.; Schechter, R. S. SOC.P e t . Ing. J . 1983, 23, 301. ( 7 ) Haque, 0.; Scamehorn, J. F. J. Dispersion Sei. Technol. 1986, 7, 129.
0 1987 American Chemical Society
