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Empirical equations of state for adsorption layer: Alkyl ethers of tri- and tetraoxyethylene glycols
Empirical Equations of State for Adsorption Layer: Alkyl Ethers of Tri- and Tetraoxyethylene Glycols S T A N I S L A W K U C H A R S K I ~ AND A D A M S O K O L O W S K I Institute of Organic and Polymer Technology, Technical University of Wroclaw, 50-370 Wroclaw, Poland
Received June 10, 1990; accepted February 21, 1991 Surface tension isotherms of n-butyl to n-octyl ethers of tri- and tetraoxyethylene glycols were approximated with orthogonal po!ynomials toget good quality values of surface pressure (II) and molar area of the adsorbed layer (a). The modified Volmer, (II,(# - ao) = Z ' , R , T), van der Waals, and virial equations of state were used to correlate II and ~ in terms of a two-dimensional gas. The determined parameters of the modified Volmer equation were used to calculate the van der Waals constant, av, and this served to evaluate the interaction energy, ev, due to the cohesion of hydrophobic chains in the adsorption layer. The interaction energy per methylene group was found to be in the range of 0.37-0.39 RT. An analogous value obtained from the second virial coefficientfor the Lennard-Jones potential was 0.14 R T. © 1991AcademicPress,Inc. INTRODUCTION Adsorption o f an amphiphile at the air-water interface leads to the formation o f the m o n o m o l e c u l a r surface layer which exhibits the surface pressure, II, II -- yo - 3',
[1]
where 3'0 is surface tension o f water and 3' is surface tension o f the solution. If interactions between molecules in the adsorption layer m a y be c o m p a r e d to those o f gases, then it is possible to describe the state o f the adsorption layer in a way similar to that o f the gases by assuming the adsorption layer to be a two-dimensional gas ( 1 - 3 ) . The plots I I - a ( ~ris m o lar area o f the adsorbed amphiphile, m 2/tool), similar to P - V plots o f gas, can be prepared by m e a s u r e m e n t o f the surface pressure with a Wilhelmy balance, or based on surface tension measurements with I I taken from Eq. [ 1]. The latter m e t h o d is suitable for the Gibbs adsorption layer formed by soluble surfactants. The spread monolayers formed by longchained molecules seem to be m o r e attractive 1To whom correspondence should be addressed.
for studies, particularly those f o r m e d by bioand polymeric materials. In this context one m a y say that the adsorbed monolayers have been forgotten. In fact, few papers were concerned with the subject. The adsorbed layer has m a i n l y the character o f a gaseous film (3) and as such it can be described by the equations derived for real gases. Deviations from an ideal surface film, behaving as a two-dimensional perfect gas, occur because the molecules are not dimensionless. The papers o f Fowkes (4), Smith ( 5 ), and Aveyard and coworkers (6, 7) show good applicability o f such equations for surfactants at the a i r / w a t e r and oil/water interfaces. The equations o f state r e c o m m e n d e d by the authors were the van der Waals, Volmer, virial, and Saraga and Prigogine (6, 8) equations o f state. A n interesting treatment o f the problem o f equations o f state o f the adsorption layer as from the point o f view o f a physicochemist was presented by Lucassen-Reynders (9), and special emphasis was put on anionic surfactants. Nevertheless, the information on the surfactants' characteristics is rather scarce and does n o t allow one to m a k e generalizations. The purpose o f this work emerged from ex-
434 0021-9797/91 $3.00 Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
435
NONIDEAL ADSORPTION FILMS
aminations of the shapes of surface tension isotherms of aqueous solutions of nonionic surfactants. Since the I I - a curves of the nonionics declined from the ideal run, it was decided to determine quantitatively how far they depart from the ideality, and to find out the correlations between the surfactant structure and the parameters describing the run of I I a curves. The surfactants in question were individual C5-C8 ethers of tri- and tetraoxyethylene glycol. They were not investigated in this context; at least, no data on the subject were found in the available literature.
3' = a + b * x + c , x 2 + d , x 3,
[2]
where x is log [ C] and C is the concentration in m o l / d m 3. Since all the surfactants in question were nonionics and the solutions were diluted, the ideality of the solutions was assumed. The polynomial approximation made by least squares yielded a 3,-log[C] relationship in the form of a mathematical expression which could be combined with the Gibbs adsorption isotherm to calculate the surface excess, I', with good accuracy within the approximation region:
MATERIALS AND METHODS
d3`
The surface tension of aqueous solutions of the nonionics was measured by the method of Wilhelmy at 293 + 0.1 K using a Pt plate. The solutions were prepared using bidistilled water. The equilibrium surface tension values were recorded when no aging was observed within 15 min. The accuracy of the method was +0.1 m N / m . The surfactants tested were the tri- and tetraoxyethylene glycol ethers of n-pentanol to n-octanol (abbreviated C x E y ) . The method of preparation and properties of the surfactants were described by the authors in ( 10, lOa). The surface tension vs concentration data were approximated using orthogonal polynomials,
- = -F,R,T dln[C] =(b+
[3]
2*c*x + 3*d*x2)/ln(lO).
The authors wrote the computer program SURFEX (11 ) to process the data from the surface tension measurements comprising polynomial approximation and calculation of adsorption parameters, i.e., II and a. Depending on the range of experimental data, 30 to 50 I I - a points were generated for further calculations. It should be noted that the use of polynomials to approximate the interfacial tension has been reported (6, 7, 12). In Table I the polynomial coefficients of the surfactants are shown as obtained by SURFEX. They are presented here for the first time.
TABLE I Values of the Polynomial Coefficients (Eq. [2]) Approximationrange as log(C) Compound
Lower
Upper
a
b
c
d
C5E3 C6E3 C7E3 C8E3 C5E4 C6E4 C7E4 C8E4
-3.564 -3.928 -4.254 -4.606 -3.902 -4.250 -4.571 -4.923
-1.559 - 1.823 -2.383 -2.895 -1.611 - 1.702 -2.265 -2.882
1.0953E+01 -9.4909E+00 -4.3295E+01 -8.3669E+01 1.1939E+01 -9.5399E+00 -3.2551E+01 -4.1004E+01
-2.9769E+01 -3.7254E+01 -5.2696E+01 -6.7681E+01 -2.6955E+01 -3.6935E+01 -4.4209E+01 -3.6148E+01
-3.5524E+00 -4.3018E+00 -6.9577E+00 -8.6297E+00 -2.9286E+00 -4.7498E+00 -5.2364E+00 - 1.3706E+00
9.5562E-03 -1.8515E-02 -2.1570E-01 -2.7237E-01 1.3199E-02 -1.3234E-01 -1.2234E-01 2.6714E-01
Journal of Colloid and Interface Science, Vol. 146,No. 2, October 15, 1991
436
KUCHARSKI AND SOKOLOWSKI RESULTS AND DISCUSSION
The I I - a curves of the surfactants studied were regular in shape, resembling a classical r u n of a P - Visotherm of gas within the range limited, arbitrarily, as the region o f applicability of the equations of a nonideal state of a two-dimensional gas. In terms of the adsorption layer, the region covers the gaseous state and eventually the extended liquid state. The limits were lower above the molar adsorption area equal to 1000 m2/mmol, i.e., 166 A2 per molecule (1 ~2 per molecule corresponds to 6.023 m 2 / m m o l ) , and upper, at surface pressure 24 m N / m . The molecule of the surfactant had 3 or 4 oxyethylene units which means 9 or 12 structure units ( /N CH2 and - O - ) in the hydrophilic part, whereas the hydrophobic part had 5 to 8 / N CH2 groups. The values of the molar adsorption area of the surfactants at saturation state showed that the area limiting factor was the polyoxyethylene chain. The residual area of C6E4, C7E4, and C8E4 was 319, 277, and 259 m2/mmol, respectively (10), and the values for the corresponding triethylene derivatives were lower by ca. 24-25 m2/mmol. For analogous alkanols the corresponding value of 150-156 m 2 / m m o l (25-26 A2) and a crosssectional area of the aliphatic chain of 115 m2/ mmol (ca. 19 ~2) were commonly assumed (2, 5).
These facts help to determine possible configurations of the surfactant molecule at the interface. It is assumed that the polyoxyethylene chain plays a basic role as a factor determining the residual area of the molecule and coils under the interface with an increase of surface pressure up to saturation of the adsorption layer. The shape of the molecule then resembles a club with a thin aliphatic chain in a vertical position above the interface. An alternative image can be a candle in a paper boat floating on water. The cohesive forces between aliphatic chains, increasing with the length of the chains, help to squeeze the coils and cause differences between surfactants. The situation when the molecule lies flat on the surface is one of low surface pressures, say below 1.5 m N / m . In this region the state of the film is often described as the "Henry's law region" and this is beyond the scope of this work. The equation of state which may be applied to describe the system studied is ( 14, 15) II,(¢-
ao) = Z ' , R , T ,
[4]
where a0 is an excluded area ( m 2 / m o l ) , a constant typical for the surfactant, and Z ' is the pseudo compressibility factor, a measure of the interaction between molecules in the film, equal to 1 when there is no adhesion and decreasing as the adhesion increases (16). If Z ' = 1, Eq. [4] becomes the Volmer equation. Equation [ 4 ] can be rewritten in the form
TABLE II Parameters of Eqs. [4] and [6] Compound
Z'
tro
Correlation coefficient Eq. [5]
av
ev R* T
C5E3 C6E3 C7E3 C8E3 C5E4 C6E4 C7E4 C8E4
0.9169 0.8238 0.7406 0.6646 1.0050 0.9585 0.8618 0.7640
223,000 215,000 201,000 190,000 242,000 223,000 218,000 209,000
0.9913 0.9897 0.9834 0.9814 0.9921 0.9870 0.9872 0.9944
1.98E+08 4.08E+08 5.71E+08 7.08E+08 - 1.27E+07 1.00E+08 3.25E+08 5.32E+08
0.364 0.779 1.166 1.529 -0.021 0.184 0.612 1.045
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
NONIDEAL ADSORPTION FILMS
437
chain length, and is lower for the triethylene glycol series as compared with tetraoxyethylene glycol derivatives. The introduction of Z ' clarifies the meaning of ao, which defined as which makes it possible to determine a0 and an excluded area approaches the cross-sec° Z ' by linear regression using the I I - a values tional area of the molecule at saturation of the produced by SURFEX. The calculated values adsorption layer. It is particularly visible in of a0 and Z ' are shown in Table II. The con- the case o f aliphatic C5-C8 alcohols. On the dition for good value parameters ao and Z ' is basis of literature data and their own meaa straight line relationship, I I , a / R , T vs 1-1. surements, the authors calculated an average In the region studied the correlation coefficient value o f a0 for these alcohols (using Eq. [ 5 ] ) of the relationship was around 0.99, making which for a comparable I I range was 119 m2/ a linear approximation justifiable (Fig. 1 ). m m o l (19.8 A2). The difference between a0 The parameters ao and Z ' are useful in and the adsorption saturation area results from characterizing the surfactants, as their values the fact that the molecules having finite dican be connected with the surfactant structure. mensions are able to cover a part of the surface. The pseudocompressibility factor Z ' decreases In the case of a circular cross section and tewith an increase in the length of the hydro- tragonal packing, a ratio of total area to the carbon surfactant chain, indicating increasing fraction covered by the adsorbed molecules is cohesion between hydrophobic parts. The ex- 1.27 and this is close to the values obtained cluded area, ao, remains correlated with the for the surfactants in question (13, 17). The parameter ao and Z ' can take on a more surfactant structure in a similar way as residual area in the saturated adsorption layer, i.e., it familiar meaning if the van der Waals equation decreases with an increase in hydrophobic is used, I I . t r _ Z ' + tro R*T ~--.-.-~*II,
[5]
qT
m./. t+0
1-
C5E4.
2-
C6E4
3 -
ideal gas
30
2 2O
o
• o o
10
I 200
J
°o
| /~00
°o o •
"e
I
e
•
-61X)
,
I ~00
I
'
m2/mMoL
FIG. 1. 1I-~r relationship of selected amphiphiles. Points as calculated by SURFEX, lines according to Eq. [41. Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
438
KUCHARSKI AND SOKOLOWSKI
Assuming ~r - a0 in Eq. [ 4 ] to have the same meaning as in Eq. [ 6 ], one obtains an expression connecting the van der Waals constant av with Z': av = I I . ~ 2 *
1 -
Z'
[7]
Z'
The unit of a~ is N . m 3/ mol. Equation [ 7 ] can be rewritten in a form where the left side, 11.(1 - Z ' ) / Z ' , is a function of 1/~ 2. The plot of this function was found to be a straight line passing through the origin. From the slope of this plot the values of av were determined (Fig. 2, Table II ), The constant a~ can be related to the interaction energy between adsorbed molecules. Assuming that the hydrophobic part of the surfactants is a stack of disks formed by CH2 groups, the disk diameter, D, and av are correlated through the potential energy of interaction, ~v, by (5, 18, 19) av = 7r*~v*D2/4.
[8]
The distance at which the disks interact is not the same as the cross-sectional area of the aliphatic chain and therefore D is determined from ao,
[9]
ao = 7r*D2/4.
[6]
(1I + av/a2)*(cr - ~o) = R * T .
Hence, E,~ = a v / ~ O .
[101
Both Z ' and ~o are affected by cohesion between the hydrophobes, which is pronounced with an increase in the length of the hydrophobes. As was mentioned previously, the coiling or recoiling of the polyoxyethylene chain is a result of cohesive interaction. The cohesion energy, Ev, can be correlated with the number of carbon atoms in the hydrophobic chain. The relationship shown in Fig. 3 may be assumed to be linear and have a slope of 0.37-0.39 R T per CH2 group. This is in good agreement with the Lennard,Jones energy parameter for the interaction of disks calculated for methane and higher hydrocarbons, which was reported to be in the range of 0.507-0.327 R T w i t h a tendency to decrease with the size of the hydrocarbon molecule (6, 21, 23). Another approach to characterize the real state of the adsorption layer was to use the virial equation of state, II*~
-
1 +-
R* T
B
+
a
C --~'
Y
6 C6E3
~'
C7[4
2 C6E~ •
2
,
,
~,
°
6
8
10
12
FIG. 2. Plot to d e t e r m i n e av. Y = 1 0 0 0 . I I ( 1 - Z ' ) / Z ' , X = 1012/cr 2. Journal of Colloid and Interface Science, Vol. 146, No: 2, October 15, 1991
X
[11]
NONIDEAL ADSORPTION FILMS Ev RT
so that the virial equation of state might be assumed to be proper for the surfactants studied. To demonstrate the contribution of each virial coefficient to the departure from ideality, the right side of Eq. [ 11 ] was taken to be an analogue of the compressibility factor, Z, the parameter which proved to be so useful in describing P - V relationships of real gases:
1,2 1,0' 0,8 0,6
O,t,, 0,20,0
5
6
8 rlc
7
FIG. 3. Relationship between ~vand number of C atoms in the hydrophobic part of the surfactant. 1; CxE3; 2, CxE4.
where B and C are the second and third virial coefficients responsible for bi- and trimolecular interactions, respectively. This is true in real gases but in the adsorption layer the interactions m a y be more complex in character. In all a s p e c t s - - n o t specifying t h e m - - t h e virial coefficients bear the departure from the ideal state of an adsorption layer. The above equation can be rewritten in the form ~,~.
1 *~=B+-~
439
[12]
and then the virial coefficients B and C can be simply obtained by a least-squares method taking I I - ~ values from SURFEX. The values of the correlation coefficients shown in Table III m a y suggest worse correlation of I I - a data as compared with Eq. [ 5 ] but in fact Fig. 4 shows a good quality data fit
Z = 1 +-
B
C + ~5.
[13]
The departure from ideality is expressed by the terms B~ a and C / a 2. The I I - a curves of the surfactants in principle lie above the ideal gas curve: I I • a = R , T signifying Z > 1. The values of C are positive and all are in the same range. The values of B are negative and not so uniform, so that the difference in the I I - a run between surfactants is incorporated in B (Fig. 5), although the total departure from ideality is in the direction of Z values > 1, i.e., the parameter C is predominant over B. The second virial coefficient is connected with bimolecular interaction and is given by
B = -½N*
(exp(-U(r)/kr) - 1),2,II,rdr,
[14]
where N is Avogadro's number, and U(r) is the intermolecular potential energy function of two adsorbed surfactant molecules whose
TABLE III Parameters of Virial Equation of State Compound
C5E3 C6E3 C7E3 C8E3 C5E4 C6E4 C7E4 C8E4
- 1.4523E+05 -2.4163E+05 -3.2899E+05 -3.9586E+05 -6.9042E+04 -9.1408E+04 -1.8521E+05 -3.1664E+05
2.4420E+ 11 2.3335E+ 11 2.1673E+11 2.0166E+ 11 2.8766E+ 11 2.3902E+ 11 2.2711E+ 11 2.3003E+ 11
Correlation coefficient Eq. [15]
~z R*T
0.9590 0.9494 0.9152 0.9077 0.9648 0.9247 0.9269 0.9745
0.508 0.645 0.787 0.900 0.391 0.432 0.568 0.752
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
440
KUCHARSKI AND SOKOLOWSKI
313
1- CSE4, 3
2 ~
"~.. 1
2-- C8E4.
o.
10
• "
i
260
I
I
!
~oo
600
u
i
i
8~)o
m~/mMoi
FIG. 4. I I - a relationship by the virial equation. Points as calculated by SURI~X, lines according to Eq.
[12], centers are separated by a distance r in the plane of the interface (20). An intermolecular potential c o m m o n l y used for gaseous, liquid, and solid states is the Lennard-Jones ( ' 6 - 1 2 ' ) potential given by U(r)=4,eL,[(~/r)
1z-(b/r)6],
[15]
where ~ and EL (which have dimensions of length and energy, respectively) are constants characteristic of the chemical species of the colliding molecules. At ~ = r the potential energy EL = 0. TO determine EL from the second virial coefficient it is convenient to introduce reduced parameters:
Z
3
c
2
B # = B/ao
[16]
T # = k* T/eL.
[17]
and
In usual cases when the model of rigid spheres is considered, the B # is calculated by dividing B by bo, which is the van der Waals I 0 constant equal to 2 , I I , N , 63 / 3, or by N , 63 f (21, 22). In our case the van der Waals conB z j -1 slant for the two-dimensional adsorption layer was ~o, so this parameter was used in Eq. [ 16 ]. 200 kO0 600 800 m2/mM01 There are correlation tables o f B # and T # from which knowing one parameter makes it posFIG. 5. Contribution of the virial coefficients to the compressibilityfactor. 1, C5E3; 2, C8E3; curves B and C sible to determine the other one. On this basis, refer to the second and third virial coefficient,respectively. the values of eL were calculated (Table III). 1
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
NONIDEAL ADSORPTION FILMS T h e plots EL VS nc are very similar to those shown in Fig. 3 (for Ev) with the slope per methylene ( m e t h y l ) g r o u p equal to 0.14 R T, m u c h lower t h a n in the case o f ev, but the absolute values o f both interaction parameters taken for the molecule as a whole are o f the same order o f magnitude. T h e e L data obtained f r o m the second virial coefficient should be treated as approximate, since B a n d C are not true virial coefficients. T h e y were obtained by simple a p p r o x i m a t i o n and bear the inaccuracy o f the a p p r o x i m a t i o n method. Nevertheless, the results are p r o m ising, so that the authors aim to write a separate paper on the virial equation o f state for the adsorption layer with m o r e numerical calculations o f the force constants. CONCLUSIONS The relationship between II and a can be approximated by the modified V o l m e r equation (Eq. [ 4 ] ). The term ( a - ~r0) introduced into the van der Waals equation makes it possible to calculate av and, hence, the interaction energy Ev, a measure o f the cohesion o f the h y d r o p h o b i c chains in the adsorption layer. The interaction energy Ev is a function o f the length o f the surfactant elements. It increases with the n u m b e r o f carbon atoms in the h y d r o p h o b e and is higher for trioxyethylene glycol surfactants as c o m p a r e d with the tetraoxyethylene glycol series. The increment o f ~v per methylene group was f o u n d to be in the range 0.37-0.39 R T . T h e virial equation o f state can also be used to evaluate the departure f r o m ideality o f the surface adsorption layer. The interaction energy per methylene group calculated from the second virial coefficient ( L e n n a r d - J o n e s potential) was in the range o f 0.14 R T , m u c h lower than that obtained from the van der Waals constant av, but the total values calculated for the surfactant molecule are o f the same order o f magnitude.
441 REFERENCES
1. Adamson, A. W., "Physical Chemistry of Surfaces," 3rd ed. Wiley, New York, 1976. 2. Schick, M. J. (ed.), "Nonionic Surfactants." Dekker, New York, 1967. 3. Birdi, K. S., "Lipids and Biopolymers at Liquid Interfaces." Plenum, New York, 1988. 4. Fowkes, F. M., J. Phys. Chem. 66, 385 (1962); 66, 1863 (1962). 5. Smith, Y., J. Colloid Interface Sci. 23, 27 (1967). 6. Aveyard, R., and Chapman, J., Can. J. Chem. 53, 916 (1975). 7. Aveyard, R., Carr, N., and Slezok, H., Can. J. Chem. 63, 2742 (1985). 8. Saraga, L., and Prigogine, I., Mem. Serv. Chim. de l'Etat 38, 109 (1953). 9. Lucassen-Reynders, E. H., "Anionic Surfactants," Dekker, New York/Basel, 1981. 10. Kucharski, S., Sokolowski,A., and Burczyk, B., Rocz. Chem. (Ann. Soc. Chim. Polonorum) 47, 2045 (1973). 10a.Sokolowski, A., and Burczyk, B., J. Colloid Interface Sci. 94, 369 (1983). 11. Kucharski, S., and Sokolowski, A., in "Proceedings, XIX Conference Spanish Committee Detergents, Granada 1988," p. 433. 12. Tomoaia, M., Ioanette, A., and Chifu, E., in "Proceedings, International Conference Colloid Surface Science (IUPAC), Budapest (1975)," Vol. I, p. 559. 13. Posner, A. M., Anderson, J. R., and Alexander, A. E., J. ColloidSci. 7, 623 (1952). 14. Schoefield, R. H., and Rideal, E. K., Proc. R. Soc. London A 109, 57 (1925). 15. Gabrielli, G., Senatra, D., Caminati, G., and Guarini, G., Colloid Polym. Sci. 266, 823 (1988). 16. Alexander, A. E., and Posner, A. M., Nature 166(No. 4219), 432 (1950). 17. Kucharski, S., and Sokolowski,A., unpublished work. 18. Hedge, D. G., J. ColloidSci. 12, 417 (1957). 19. Reiss, H., Frisch, H. L., and Lebowitz, J. L., J. Chem. Phys. 34, 369 (1959). 20. Stigter, D., and Dill, K. A. Langmuir 2, 791 (1986). 21. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., "Molecular Theory of Gases and Liquids," Wiley, New York, 1954. 22. Mason, E. A., and Spurling, T. H., "The Virial Equation of State." Mir, Moscow, 1972 [Russian translation]. 23. Cholinski, J., Szafranski, A., and Wyrzykowska-Stankiewicz, D., "Computer Aided Second Virial Coefficient Data for Organic Individual Compounds and Binary Systems." PWN, Warsaw, 1986.
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
Received June 10, 1990; accepted February 21, 1991 Surface tension isotherms of n-butyl to n-octyl ethers of tri- and tetraoxyethylene glycols were approximated with orthogonal po!ynomials toget good quality values of surface pressure (II) and molar area of the adsorbed layer (a). The modified Volmer, (II,(# - ao) = Z ' , R , T), van der Waals, and virial equations of state were used to correlate II and ~ in terms of a two-dimensional gas. The determined parameters of the modified Volmer equation were used to calculate the van der Waals constant, av, and this served to evaluate the interaction energy, ev, due to the cohesion of hydrophobic chains in the adsorption layer. The interaction energy per methylene group was found to be in the range of 0.37-0.39 RT. An analogous value obtained from the second virial coefficientfor the Lennard-Jones potential was 0.14 R T. © 1991AcademicPress,Inc. INTRODUCTION Adsorption o f an amphiphile at the air-water interface leads to the formation o f the m o n o m o l e c u l a r surface layer which exhibits the surface pressure, II, II -- yo - 3',
[1]
where 3'0 is surface tension o f water and 3' is surface tension o f the solution. If interactions between molecules in the adsorption layer m a y be c o m p a r e d to those o f gases, then it is possible to describe the state o f the adsorption layer in a way similar to that o f the gases by assuming the adsorption layer to be a two-dimensional gas ( 1 - 3 ) . The plots I I - a ( ~ris m o lar area o f the adsorbed amphiphile, m 2/tool), similar to P - V plots o f gas, can be prepared by m e a s u r e m e n t o f the surface pressure with a Wilhelmy balance, or based on surface tension measurements with I I taken from Eq. [ 1]. The latter m e t h o d is suitable for the Gibbs adsorption layer formed by soluble surfactants. The spread monolayers formed by longchained molecules seem to be m o r e attractive 1To whom correspondence should be addressed.
for studies, particularly those f o r m e d by bioand polymeric materials. In this context one m a y say that the adsorbed monolayers have been forgotten. In fact, few papers were concerned with the subject. The adsorbed layer has m a i n l y the character o f a gaseous film (3) and as such it can be described by the equations derived for real gases. Deviations from an ideal surface film, behaving as a two-dimensional perfect gas, occur because the molecules are not dimensionless. The papers o f Fowkes (4), Smith ( 5 ), and Aveyard and coworkers (6, 7) show good applicability o f such equations for surfactants at the a i r / w a t e r and oil/water interfaces. The equations o f state r e c o m m e n d e d by the authors were the van der Waals, Volmer, virial, and Saraga and Prigogine (6, 8) equations o f state. A n interesting treatment o f the problem o f equations o f state o f the adsorption layer as from the point o f view o f a physicochemist was presented by Lucassen-Reynders (9), and special emphasis was put on anionic surfactants. Nevertheless, the information on the surfactants' characteristics is rather scarce and does n o t allow one to m a k e generalizations. The purpose o f this work emerged from ex-
434 0021-9797/91 $3.00 Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
435
NONIDEAL ADSORPTION FILMS
aminations of the shapes of surface tension isotherms of aqueous solutions of nonionic surfactants. Since the I I - a curves of the nonionics declined from the ideal run, it was decided to determine quantitatively how far they depart from the ideality, and to find out the correlations between the surfactant structure and the parameters describing the run of I I a curves. The surfactants in question were individual C5-C8 ethers of tri- and tetraoxyethylene glycol. They were not investigated in this context; at least, no data on the subject were found in the available literature.
3' = a + b * x + c , x 2 + d , x 3,
[2]
where x is log [ C] and C is the concentration in m o l / d m 3. Since all the surfactants in question were nonionics and the solutions were diluted, the ideality of the solutions was assumed. The polynomial approximation made by least squares yielded a 3,-log[C] relationship in the form of a mathematical expression which could be combined with the Gibbs adsorption isotherm to calculate the surface excess, I', with good accuracy within the approximation region:
MATERIALS AND METHODS
d3`
The surface tension of aqueous solutions of the nonionics was measured by the method of Wilhelmy at 293 + 0.1 K using a Pt plate. The solutions were prepared using bidistilled water. The equilibrium surface tension values were recorded when no aging was observed within 15 min. The accuracy of the method was +0.1 m N / m . The surfactants tested were the tri- and tetraoxyethylene glycol ethers of n-pentanol to n-octanol (abbreviated C x E y ) . The method of preparation and properties of the surfactants were described by the authors in ( 10, lOa). The surface tension vs concentration data were approximated using orthogonal polynomials,
- = -F,R,T dln[C] =(b+
[3]
2*c*x + 3*d*x2)/ln(lO).
The authors wrote the computer program SURFEX (11 ) to process the data from the surface tension measurements comprising polynomial approximation and calculation of adsorption parameters, i.e., II and a. Depending on the range of experimental data, 30 to 50 I I - a points were generated for further calculations. It should be noted that the use of polynomials to approximate the interfacial tension has been reported (6, 7, 12). In Table I the polynomial coefficients of the surfactants are shown as obtained by SURFEX. They are presented here for the first time.
TABLE I Values of the Polynomial Coefficients (Eq. [2]) Approximationrange as log(C) Compound
Lower
Upper
a
b
c
d
C5E3 C6E3 C7E3 C8E3 C5E4 C6E4 C7E4 C8E4
-3.564 -3.928 -4.254 -4.606 -3.902 -4.250 -4.571 -4.923
-1.559 - 1.823 -2.383 -2.895 -1.611 - 1.702 -2.265 -2.882
1.0953E+01 -9.4909E+00 -4.3295E+01 -8.3669E+01 1.1939E+01 -9.5399E+00 -3.2551E+01 -4.1004E+01
-2.9769E+01 -3.7254E+01 -5.2696E+01 -6.7681E+01 -2.6955E+01 -3.6935E+01 -4.4209E+01 -3.6148E+01
-3.5524E+00 -4.3018E+00 -6.9577E+00 -8.6297E+00 -2.9286E+00 -4.7498E+00 -5.2364E+00 - 1.3706E+00
9.5562E-03 -1.8515E-02 -2.1570E-01 -2.7237E-01 1.3199E-02 -1.3234E-01 -1.2234E-01 2.6714E-01
Journal of Colloid and Interface Science, Vol. 146,No. 2, October 15, 1991
436
KUCHARSKI AND SOKOLOWSKI RESULTS AND DISCUSSION
The I I - a curves of the surfactants studied were regular in shape, resembling a classical r u n of a P - Visotherm of gas within the range limited, arbitrarily, as the region o f applicability of the equations of a nonideal state of a two-dimensional gas. In terms of the adsorption layer, the region covers the gaseous state and eventually the extended liquid state. The limits were lower above the molar adsorption area equal to 1000 m2/mmol, i.e., 166 A2 per molecule (1 ~2 per molecule corresponds to 6.023 m 2 / m m o l ) , and upper, at surface pressure 24 m N / m . The molecule of the surfactant had 3 or 4 oxyethylene units which means 9 or 12 structure units ( /N CH2 and - O - ) in the hydrophilic part, whereas the hydrophobic part had 5 to 8 / N CH2 groups. The values of the molar adsorption area of the surfactants at saturation state showed that the area limiting factor was the polyoxyethylene chain. The residual area of C6E4, C7E4, and C8E4 was 319, 277, and 259 m2/mmol, respectively (10), and the values for the corresponding triethylene derivatives were lower by ca. 24-25 m2/mmol. For analogous alkanols the corresponding value of 150-156 m 2 / m m o l (25-26 A2) and a crosssectional area of the aliphatic chain of 115 m2/ mmol (ca. 19 ~2) were commonly assumed (2, 5).
These facts help to determine possible configurations of the surfactant molecule at the interface. It is assumed that the polyoxyethylene chain plays a basic role as a factor determining the residual area of the molecule and coils under the interface with an increase of surface pressure up to saturation of the adsorption layer. The shape of the molecule then resembles a club with a thin aliphatic chain in a vertical position above the interface. An alternative image can be a candle in a paper boat floating on water. The cohesive forces between aliphatic chains, increasing with the length of the chains, help to squeeze the coils and cause differences between surfactants. The situation when the molecule lies flat on the surface is one of low surface pressures, say below 1.5 m N / m . In this region the state of the film is often described as the "Henry's law region" and this is beyond the scope of this work. The equation of state which may be applied to describe the system studied is ( 14, 15) II,(¢-
ao) = Z ' , R , T ,
[4]
where a0 is an excluded area ( m 2 / m o l ) , a constant typical for the surfactant, and Z ' is the pseudo compressibility factor, a measure of the interaction between molecules in the film, equal to 1 when there is no adhesion and decreasing as the adhesion increases (16). If Z ' = 1, Eq. [4] becomes the Volmer equation. Equation [ 4 ] can be rewritten in the form
TABLE II Parameters of Eqs. [4] and [6] Compound
Z'
tro
Correlation coefficient Eq. [5]
av
ev R* T
C5E3 C6E3 C7E3 C8E3 C5E4 C6E4 C7E4 C8E4
0.9169 0.8238 0.7406 0.6646 1.0050 0.9585 0.8618 0.7640
223,000 215,000 201,000 190,000 242,000 223,000 218,000 209,000
0.9913 0.9897 0.9834 0.9814 0.9921 0.9870 0.9872 0.9944
1.98E+08 4.08E+08 5.71E+08 7.08E+08 - 1.27E+07 1.00E+08 3.25E+08 5.32E+08
0.364 0.779 1.166 1.529 -0.021 0.184 0.612 1.045
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
NONIDEAL ADSORPTION FILMS
437
chain length, and is lower for the triethylene glycol series as compared with tetraoxyethylene glycol derivatives. The introduction of Z ' clarifies the meaning of ao, which defined as which makes it possible to determine a0 and an excluded area approaches the cross-sec° Z ' by linear regression using the I I - a values tional area of the molecule at saturation of the produced by SURFEX. The calculated values adsorption layer. It is particularly visible in of a0 and Z ' are shown in Table II. The con- the case o f aliphatic C5-C8 alcohols. On the dition for good value parameters ao and Z ' is basis of literature data and their own meaa straight line relationship, I I , a / R , T vs 1-1. surements, the authors calculated an average In the region studied the correlation coefficient value o f a0 for these alcohols (using Eq. [ 5 ] ) of the relationship was around 0.99, making which for a comparable I I range was 119 m2/ a linear approximation justifiable (Fig. 1 ). m m o l (19.8 A2). The difference between a0 The parameters ao and Z ' are useful in and the adsorption saturation area results from characterizing the surfactants, as their values the fact that the molecules having finite dican be connected with the surfactant structure. mensions are able to cover a part of the surface. The pseudocompressibility factor Z ' decreases In the case of a circular cross section and tewith an increase in the length of the hydro- tragonal packing, a ratio of total area to the carbon surfactant chain, indicating increasing fraction covered by the adsorbed molecules is cohesion between hydrophobic parts. The ex- 1.27 and this is close to the values obtained cluded area, ao, remains correlated with the for the surfactants in question (13, 17). The parameter ao and Z ' can take on a more surfactant structure in a similar way as residual area in the saturated adsorption layer, i.e., it familiar meaning if the van der Waals equation decreases with an increase in hydrophobic is used, I I . t r _ Z ' + tro R*T ~--.-.-~*II,
[5]
qT
m./. t+0
1-
C5E4.
2-
C6E4
3 -
ideal gas
30
2 2O
o
• o o
10
I 200
J
°o
| /~00
°o o •
"e
I
e
•
-61X)
,
I ~00
I
'
m2/mMoL
FIG. 1. 1I-~r relationship of selected amphiphiles. Points as calculated by SURFEX, lines according to Eq. [41. Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
438
KUCHARSKI AND SOKOLOWSKI
Assuming ~r - a0 in Eq. [ 4 ] to have the same meaning as in Eq. [ 6 ], one obtains an expression connecting the van der Waals constant av with Z': av = I I . ~ 2 *
1 -
Z'
[7]
Z'
The unit of a~ is N . m 3/ mol. Equation [ 7 ] can be rewritten in a form where the left side, 11.(1 - Z ' ) / Z ' , is a function of 1/~ 2. The plot of this function was found to be a straight line passing through the origin. From the slope of this plot the values of av were determined (Fig. 2, Table II ), The constant a~ can be related to the interaction energy between adsorbed molecules. Assuming that the hydrophobic part of the surfactants is a stack of disks formed by CH2 groups, the disk diameter, D, and av are correlated through the potential energy of interaction, ~v, by (5, 18, 19) av = 7r*~v*D2/4.
[8]
The distance at which the disks interact is not the same as the cross-sectional area of the aliphatic chain and therefore D is determined from ao,
[9]
ao = 7r*D2/4.
[6]
(1I + av/a2)*(cr - ~o) = R * T .
Hence, E,~ = a v / ~ O .
[101
Both Z ' and ~o are affected by cohesion between the hydrophobes, which is pronounced with an increase in the length of the hydrophobes. As was mentioned previously, the coiling or recoiling of the polyoxyethylene chain is a result of cohesive interaction. The cohesion energy, Ev, can be correlated with the number of carbon atoms in the hydrophobic chain. The relationship shown in Fig. 3 may be assumed to be linear and have a slope of 0.37-0.39 R T per CH2 group. This is in good agreement with the Lennard,Jones energy parameter for the interaction of disks calculated for methane and higher hydrocarbons, which was reported to be in the range of 0.507-0.327 R T w i t h a tendency to decrease with the size of the hydrocarbon molecule (6, 21, 23). Another approach to characterize the real state of the adsorption layer was to use the virial equation of state, II*~
-
1 +-
R* T
B
+
a
C --~'
Y
6 C6E3
~'
C7[4
2 C6E~ •
2
,
,
~,
°
6
8
10
12
FIG. 2. Plot to d e t e r m i n e av. Y = 1 0 0 0 . I I ( 1 - Z ' ) / Z ' , X = 1012/cr 2. Journal of Colloid and Interface Science, Vol. 146, No: 2, October 15, 1991
X
[11]
NONIDEAL ADSORPTION FILMS Ev RT
so that the virial equation of state might be assumed to be proper for the surfactants studied. To demonstrate the contribution of each virial coefficient to the departure from ideality, the right side of Eq. [ 11 ] was taken to be an analogue of the compressibility factor, Z, the parameter which proved to be so useful in describing P - V relationships of real gases:
1,2 1,0' 0,8 0,6
O,t,, 0,20,0
5
6
8 rlc
7
FIG. 3. Relationship between ~vand number of C atoms in the hydrophobic part of the surfactant. 1; CxE3; 2, CxE4.
where B and C are the second and third virial coefficients responsible for bi- and trimolecular interactions, respectively. This is true in real gases but in the adsorption layer the interactions m a y be more complex in character. In all a s p e c t s - - n o t specifying t h e m - - t h e virial coefficients bear the departure from the ideal state of an adsorption layer. The above equation can be rewritten in the form ~,~.
1 *~=B+-~
439
[12]
and then the virial coefficients B and C can be simply obtained by a least-squares method taking I I - ~ values from SURFEX. The values of the correlation coefficients shown in Table III m a y suggest worse correlation of I I - a data as compared with Eq. [ 5 ] but in fact Fig. 4 shows a good quality data fit
Z = 1 +-
B
C + ~5.
[13]
The departure from ideality is expressed by the terms B~ a and C / a 2. The I I - a curves of the surfactants in principle lie above the ideal gas curve: I I • a = R , T signifying Z > 1. The values of C are positive and all are in the same range. The values of B are negative and not so uniform, so that the difference in the I I - a run between surfactants is incorporated in B (Fig. 5), although the total departure from ideality is in the direction of Z values > 1, i.e., the parameter C is predominant over B. The second virial coefficient is connected with bimolecular interaction and is given by
B = -½N*
(exp(-U(r)/kr) - 1),2,II,rdr,
[14]
where N is Avogadro's number, and U(r) is the intermolecular potential energy function of two adsorbed surfactant molecules whose
TABLE III Parameters of Virial Equation of State Compound
C5E3 C6E3 C7E3 C8E3 C5E4 C6E4 C7E4 C8E4
- 1.4523E+05 -2.4163E+05 -3.2899E+05 -3.9586E+05 -6.9042E+04 -9.1408E+04 -1.8521E+05 -3.1664E+05
2.4420E+ 11 2.3335E+ 11 2.1673E+11 2.0166E+ 11 2.8766E+ 11 2.3902E+ 11 2.2711E+ 11 2.3003E+ 11
Correlation coefficient Eq. [15]
~z R*T
0.9590 0.9494 0.9152 0.9077 0.9648 0.9247 0.9269 0.9745
0.508 0.645 0.787 0.900 0.391 0.432 0.568 0.752
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
440
KUCHARSKI AND SOKOLOWSKI
313
1- CSE4, 3
2 ~
"~.. 1
2-- C8E4.
o.
10
• "
i
260
I
I
!
~oo
600
u
i
i
8~)o
m~/mMoi
FIG. 4. I I - a relationship by the virial equation. Points as calculated by SURI~X, lines according to Eq.
[12], centers are separated by a distance r in the plane of the interface (20). An intermolecular potential c o m m o n l y used for gaseous, liquid, and solid states is the Lennard-Jones ( ' 6 - 1 2 ' ) potential given by U(r)=4,eL,[(~/r)
1z-(b/r)6],
[15]
where ~ and EL (which have dimensions of length and energy, respectively) are constants characteristic of the chemical species of the colliding molecules. At ~ = r the potential energy EL = 0. TO determine EL from the second virial coefficient it is convenient to introduce reduced parameters:
Z
3
c
2
B # = B/ao
[16]
T # = k* T/eL.
[17]
and
In usual cases when the model of rigid spheres is considered, the B # is calculated by dividing B by bo, which is the van der Waals I 0 constant equal to 2 , I I , N , 63 / 3, or by N , 63 f (21, 22). In our case the van der Waals conB z j -1 slant for the two-dimensional adsorption layer was ~o, so this parameter was used in Eq. [ 16 ]. 200 kO0 600 800 m2/mM01 There are correlation tables o f B # and T # from which knowing one parameter makes it posFIG. 5. Contribution of the virial coefficients to the compressibilityfactor. 1, C5E3; 2, C8E3; curves B and C sible to determine the other one. On this basis, refer to the second and third virial coefficient,respectively. the values of eL were calculated (Table III). 1
Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991
NONIDEAL ADSORPTION FILMS T h e plots EL VS nc are very similar to those shown in Fig. 3 (for Ev) with the slope per methylene ( m e t h y l ) g r o u p equal to 0.14 R T, m u c h lower t h a n in the case o f ev, but the absolute values o f both interaction parameters taken for the molecule as a whole are o f the same order o f magnitude. T h e e L data obtained f r o m the second virial coefficient should be treated as approximate, since B a n d C are not true virial coefficients. T h e y were obtained by simple a p p r o x i m a t i o n and bear the inaccuracy o f the a p p r o x i m a t i o n method. Nevertheless, the results are p r o m ising, so that the authors aim to write a separate paper on the virial equation o f state for the adsorption layer with m o r e numerical calculations o f the force constants. CONCLUSIONS The relationship between II and a can be approximated by the modified V o l m e r equation (Eq. [ 4 ] ). The term ( a - ~r0) introduced into the van der Waals equation makes it possible to calculate av and, hence, the interaction energy Ev, a measure o f the cohesion o f the h y d r o p h o b i c chains in the adsorption layer. The interaction energy Ev is a function o f the length o f the surfactant elements. It increases with the n u m b e r o f carbon atoms in the h y d r o p h o b e and is higher for trioxyethylene glycol surfactants as c o m p a r e d with the tetraoxyethylene glycol series. The increment o f ~v per methylene group was f o u n d to be in the range 0.37-0.39 R T . T h e virial equation o f state can also be used to evaluate the departure f r o m ideality o f the surface adsorption layer. The interaction energy per methylene group calculated from the second virial coefficient ( L e n n a r d - J o n e s potential) was in the range o f 0.14 R T , m u c h lower than that obtained from the van der Waals constant av, but the total values calculated for the surfactant molecule are o f the same order o f magnitude.
441 REFERENCES
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Journal of Colloid and Interface Science, Vol. 146, No. 2, October 15, 1991