15478
J. Phys. Chem. 1996, 100, 15478-15482
Equation of State for Insoluble Monolayers of Aggregating Amphiphilic Molecules V. B. Fainerman,†,‡ D. Vollhardt,*,† and V. Melzer† Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Rudower Chaussee 5, D-12 489 Berlin, Germany, and Institute of Technical Ecology, Bul. SheVchenko 25, Donetsk 340017, Ukraine ReceiVed: February 21, 1996; In Final Form: April 30, 1996X
For Langmuir monolayers at the air-water interface, an equation of state is theoretically derived which describes the main phase transition between the gaslike and the condensed phases. The theoretical treatment considers the formation of two-dimensional aggregates and describes the nonhorizontal phase transition of the surface pressure-area isotherms and its dependence on the temperature. The equilibrium between the aggregates and monomers was treated using Buttler’s equation for the chemical potential of monomers and aggregates within the surface. The results predicted by the theory agree well with the experimental surface pressurearea isotherms over a large temperature range.
Introduction
Π)
Amphiphilic monolayers as representative two-dimensional model systems have been the subject of numerous investigations. In recent years, sensitive optical microscopy, e.g., Brewster angle microscopy, has substantiated a wide variety of condensed phase structures formed in the first-order two-phase coexistence region.1-6 The main feature of the two-dimensional equation of state or pressure-area (π-A) isotherm is the plateau region which attributes the main transition from a fluid phase of low density to a condensed phase. For this first-order phase transition, the most commonly used two-dimensional equations of state predict horizontal lines in the π-A isotherm. However, the experimental π-A isotherms are characterized by nonhorizontal lines.1-3,7-9 The slope of these straight lines usually increases with temperature.2,3 In spite of the variety of the condensed phase domains, the size of these aggregates can be estimated to range from a few micrometers to tens and hundreds of micrometers, even at the beginning of the two-phase transition. This corresponds to aggregation numbers of the order of 106 and higher. For such high values, the application of the first-order phase transition model seems to be justified from the theoretical point of view. This model, however predicts that the pressure dependence on the area per molecule is represented by a horizontal line.10,11 Recently, some attempts were made to resolve this discrepancy.12-14 The theoretical results obtained for the pressure dependence on the area per molecule were in qualitative, not quantitative, agreement with the experimental data. The nonhorizontality of the straight line of the “plateau” region was predicted for unrealistically low values of the aggregation number (less than 1000).14 In the present study the equation of state is derived for Langmuir monolayers considering the formation of twodimensional aggregates in the course of which the surface pressure is affected by aggregates and monomers. Moreover, the influence of small aggregates formed in the gas-analogous fluid phase of the monolayer on surface pressure is estimated. Theory Gaseous insoluble monolayers of amphiphiles are satisfactorily described by the commonly used 2D equation of state †
Max-Planck-Institut fu¨r Kolloid- und Grenzfla¨chenforschung. ‡ Institute of Technical Ecology. X Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)00523-0 CCC: $12.00
kT -B A-ω
(1)
where Π is the surface pressure, k the Boltzmann constant, T the temperature, A the area per molecule in the monolayer, and ω the net molecule area, i.e., the area actually occupied by the monolayer molecules. For B ) const ) Π* (cohesion pressure), eq 1 reduces to Volmer’s equation,15 while for B ) a/A2 (where a is a constant) eq 1 is transformed into the 2D van der Waals equation. Other forms of the equation of state for the gaseous monolayer have also been proposed.14,16-18 This particular form of eq 1 was chosen here because it agrees well with the experimental data. The main purpose of the present work is to show that the equation of state for the gaseous monolayer can also be extended to further monolayer stages in which an additional condensed phase is formed owing to two-dimensional aggregation. This means a first-order phase transition must be considered where two thermodynamic phases coexist. In this respect it is necessary to introduce into this equation corrections concerning the molecular aggregation. In addition to eq 1, any other equations of state suitable for the description of gaseous monolayers can be extended to describe the formation of an additional condensed phase due to two-dimensional aggregation. For example, the equation of state for the monolayer in the two-dimensional phase transition region can be derived from Buttler’s theory.19 The resulting expression, however, is of a logarithmic type similar to that obtained in ref 14 and cannot satisfactorily describe the experimental data obtained for insoluble monolayers both in the region of the gaseous monolayer and in the two-dimensional phase transition region. If the area per molecule, A, and net molecule area, ω, as described in this equation, are equated to corresponding values of mean areas for all kinetic entities in the monolayer, eq 1 can also be regarded as the equation of state in the two-phase coexistence region. Here both monomers and aggregates (nmers) are referred to as kinetic entities. Consider aggregates with an agggregation number n. If there are m such aggregates per monomer in the surface, the mean area per kinetic entity, Ak, can be expressed via the area per molecule, A, in the monolayer by
Ak )
A(1 + mn) 1+m
and similarly for the mean net area of kinetic entity © 1996 American Chemical Society
(2)
Equation of State for Insoluble Monolayers
ωk )
J. Phys. Chem., Vol. 100, No. 38, 1996 15479
ω(1 + mn) 1+m
(3)
where ω is the net area value actually occupied by a monomer molecule and in general m , 1. Here it is assumed that the value of ω for free molecules is equal to that for the molecules in the aggregate. Introducing expressions 2 and 3 for the corresponding area of the kinetic entities into eq 1, one obtains
Π)
kT -B (A - ω)K
(4)
where K ) (1 + mn)/(1 + m). Since n . 1 and we have m , 1, K may be approximated as K ) 1 + mn. It can be clearly seen that K > 1: Note that this approximation means that if aggregates are formed, the surface tension is lower than it would be for the same value of A in a gaseous monolayer. The expressions for the constants K and m can be obtained from the monomer-aggregate equilibrium condition in the monolayer. The chemical potential of monomeric molecules in a surface layer is given by Buttler’s equation19 as
µ1 ) µ°1 + kT ln(f1X1) - σω1
(5)
where µ°1 is the standard value for chemical potential, σ the surface tension, f1 the activity coefficient, X1 the mole fraction of monomers in the surface, and ω1 ) ω, the area of the monomeric molecule. For the n-mers a similar equation holds
µn ) µ°n + kT ln(fnXn) - σωn
(7)
where Kn ) exp[(nµ°1 - µ°n)/kT] is a two-dimensional aggregation constant. Next we introduce the surface concentration (adsorption) of monomers and aggregates into eq 7 instead of the mole fraction values where the mole fraction of the ith component in the surface layer is defined as
Xi ) Γi/∑Γi
(8)
The summation in eq 8 is performed over all components including the solvent. The dividing surface has been chosen according to Lucassen-Reynders20,21 so that the relation ∑Γi ) 1/ωΣ holds, where ωΣ is the mean partial molar surface area of the monomers. In our case
ωΣ )
ωiΓi ∑ 1,n Γi ∑ 1,n
≈
ω1Γ1 + nω1Γn Γ1 + nΓn
≈ ω1
Kn )
ΓnωΣ
)
n
(Γ1ωΣ)
Γn Γn1
ωΣ-(n-1)
(10)
Next the critical surface concentration of the aggregate formation (two-dimensional phase transition) Γc1 is defined as the surface monomer concentration for which Γn ) Γ1, where is a small quantity. The constant Kn can be expressed via the Γc1 as
ω-(n-1) Kn ) Γ-(n-1) c1 1
(11)
Substituting this into eq 10, we now obtain
()
Γn ) Γ1
Γ1 Γc
n-1
(12)
where Γc is the reduced critical surface concentration (adsorption) value and is expressed as follows
(6)
and thus for equilibrium µn ) nµ1. It can be assumed that f1 ) fn ) 1 and ωn ) nω1. Then upon multiplication of eq 5 by n and comparison of it with eq 6, one obtains
Kn ) Xn/Xn1
given surfactant (monomer or aggregate) in the surface is equal to the surface fraction occupied by this surfactant. LucassenReynders’ condition can alternatively be represented as ωΣ ) ω0, where ω0 is the partial molecular area of the solvent.20,21 It is easily seen that the definition of eq 8 under the condition ∑Γi ) 1/ω1 is also equivalent to the choice of the dividing surface for which the solute molecules are not included into the surface layer, but the sum of adsorption sites is fixed. This is just the case for insoluble monolayer. Using eq 8, eq 7 is transformed into
Γc ) Γc11/(n-1)(ω1/ωΣ)n-1
(13)
Equation 13 can be simplified to Γc ≈ Γc1, when ω1 ≈ ωΣ and n > 1000 (in this case 1/(n-1) ≈ 1 for ≈ 0.001). It follows from eq 12 that Γn ) 0 for Γ1 < Γc , and Γn > 0 for Γ1 + Γn > Γ c. The total number of monomers per unit surface is equal to
Γ ) Γ1 + nΓn ) Γ1[1 + n(Γ1/Γc)n-1]
(14)
and the concentration of the kinetic entities Γk is given by the relation
Γk ) Γ1 + Γn ) Γ1[1 + (Γ1/Γc)n-1]
(15)
Introducing Γ1 from eq 14 into eq 15, we obtain the corresponding expression for the area per molecule (kinetic entity)
1 + n(Ac/A1)n-1
Ak ) A
1 + (Ac/A1)n-1
) KA
(16)
It can be seen that eq 16 can be reduced to eq 3 for m ) (Ac/ A1)n-1. This expression for m follows also directly from eq 12
(9)
The approximate equality in eq 9 indicates that it is assumed that the net surface of the monomer outside the aggregate is equal to its surface within the aggregate. The reasons for the choice of the dividing surface according to Lucassen-Reynders were discussed by us in detail recently.22 Here only the main points will be emphasized, namely: (i) eq 5 and its analogue for the solvent are reduced to Szyszkowski’s equation in the limiting case of one surfactant, and (ii) the mole fraction of a
m)
() ()
Γn Γ1 ) Γ1 Γc
n-1
)
Ac A1
n-1
(17)
Equation 14 can be expressed in terms of area per molecule as follows
A ) A1[1 + n(Ac/A1)n-1]-1
(18)
Equation (18) specifies the dependence of the surface area per monomer, A1, on the total molecule surface A of the monolayer in the two-phase coexistence region transition, i.e., for A < Ac.
15480 J. Phys. Chem., Vol. 100, No. 38, 1996
Fainerman et al.
Figure 1. Dependence of the coefficient K on A/Ac for different aggregation numbers n.
Using the formulae 16-18, the values of the coefficients K and m and the values A1 were calculated as functions of A/Ac for n ) 103-106. From the results of these calculations the following conclusions are drawn: (1) In the region A < Ac, the area, A1, occupied by a monomer does not vary significantly and is equal to the molecular area in the critical point, A1 ≈ Ac, with an accuracy of not less than 10/n. (2) Τhe value of the coefficient m is m , 1 (usually m < 10/n), and thus K ≈ 1 + mn. (3) The value of the coefficient K for n > 103 does not depend on the aggregation number (see Figure 1), and depends only on A/Ac. Using eq 18, the following expression for the coefficient K for A < Ac and n > 103 is obtained when as before K approximates to K ) 1 + mn which can be stated as
K ) 1 + mn ) A1/A ≈ Ac/A
(19)
The results of the calculations of K by using the exact eq 16 are presented in Figure 1 and coincide with the data obtained from the approximate relation 19. Introducing eq 19 into formula 4 leads to the equation of state for insoluble monolayers with aggregating molecules
Π)
kT -B (A - ω)(AcA)
(20)
Equation 20 is valid for A e Ac. For A > Ac, it follows that m ) 0 and K ) 1. In the derivation of eq 7, the approximation ωn ) nω1 was used. For the case that the net area of monomer in the aggregate differs from the net area of the nonaggregated monomer, the coefficient K is obtained from eqs 5 and 6
σ(nω1 - ωn) kT
K ) K0 exp
(21)
where K0 ) Ac/A (for n > 103) or is defined by eq 16. The reason for the nonideality of the gaseous monolayer, which made it necessary to introduce the coefficient B into eqs 1, 4, and 20, is the formation of small aggregates in the region A > Ac. The aggregation of adsorbed organic molecules to cluster sizes up to approximately 20 molecules was also concluded by modeling of adsorption isotherms using the Ising lattice gas model.23 For the case that the condition n . 1 is
Figure 2. Dependence of the coefficient K1 on A/Ac for different aggregation numbers n.
not satisfied, the constant Kn in eq 10 can be expressed via the surface concentration of monomers Γ1/2, for which the number of monomers per unit surface is equal to the number of aggregates, i.e., Γn ) Γ1 ) Γ1/2. It follows then that -(n-1) -(n-1) Kn ) Γ1/2 ω1
(22)
For small values of n, using the relation 22, one can rewrite eq 18 as
A ) A1[1 + n(A1/2/A1)n-1]-1
(23)
The dependence of the coefficient K1 ) (1 + mn)/(1 + m) on the ratio A/A1/2 for the values n ) 2-15 is shown in Figure 2. In contrast to the behavior of K for n > 103 (see Figure 1), for small n the value of K1 depends strongly on n. Now the situation when nω1 * ωn is considered. By omitting the B term in eq 1, the equation for gaseous monolayers in which small aggregates are formed reduces to
Π)
kT (A - ω)K1
(24)
where K1 ) K* exp(-Π∆ω/kT), ∆ω ) nω1 - ωn, and K* is the value of the coefficient K1 ) (1 + mn)/(1 + m) for Π ) 0. Using relation 24, eq 20 can be transformed into
Π)
kT K1(A - ω)(Ac/A)
(25)
The coefficient K1 in eq 25 can be expressed via the critical point coordinates due to the weak dependence of Π on A in the region of two-dimensional phase transition. Therefore
Π)
Πc(Ac - ω) Ac(1 - ω/A)
(26)
which is valid in the region A e Ac, where Πc and Ac are the coordinates of the critical point on the pressure vs area isotherm, at which the two-dimensional phase transition begins. It follows from eq 26 that in this region the value of Π does not remain constant and increases with the decrease of A. The slope of the Π - A curve increases with the decrease of the Ac value, which agrees well with published experimental data. The shape
Equation of State for Insoluble Monolayers
J. Phys. Chem., Vol. 100, No. 38, 1996 15481
TABLE 1: Parameters of the Phase Transition of 1-Monostearoyl-rac-glycerol T Ac (nm2/ Πc Π* K* × ωf (nm2/ (°C) molecule) (mN/m) molecule) (mN/m) exp(-(Πc∆ω)/(kT)) 23 30 35 40 45
0.67 0.64 0.55 0.48 0.44
0.2 1.4 4 7.7 11.9
0.259 0.248 0.222 0.203 0.184
8.88 8.56 8.88 8.91 8.05
6.79 3.22 2.16 1.68
TABLE 2: Parameters of Phase Transition of 1-Monopalmitoyl-rac-glycerol T Ac (nm2/ Πc Π* K* × ωf (nm2/ (°C) molecule) (mN/m) molecule) (mN/m) exp(-(Πc∆ω)/(kT)) 15 17 20 23 25 27 30 35 42
0.64 0.60 0.55 0.50 0.47 0.445 0.41 0.37 0.33
0.8 1.8 3.7 5.9 8.0 9.9 13.1 18.0 26.6
0.25 0.23 0.20 0.15 0.14 0.15 0.13 0.13 0.10
8.66 8.73 8.55 8.69 8.45 8.5 8.91 10.34 12.9
11.80 5.85 3.31 2.47 2.06 1.86 1.68 1.57 1.49
Figure 3. Dependence of surface pressure on the area per molecule for 1-monostearoyl-rac-glycerol. Experimental results5 are presented by curves; symbols correspond to the calculations using eq 26: open symbols, ω ) 0.22 nm2; filled symbols, ω ) ωf (Table 1).
of the Π - A curve predicted by eq 26 does not depend on the aggregation number value for large n and characterizes realistic n values of the order of 103-106 and higher. Results and Discussion To verify the monolayer equations of state which have been derived in this work, experimental data recently obtained for the equilibrium surface pressure dependence on the area for 1-monostearoyl-rac-glycerol5 (MSRG) and 1-monopalmitoylrac-glycerol6 (MPRG) over a wide temperature range were used. The critical point characteristics (Πc and Ac) and other parameters involved in the equations of state 1, 24, and 26 are presented in Tables 1and 2. Figure 3 shows both experimental data and theoretical curves Π vs A for MSRG monolayers in two-dimensional phase transition region, while the same dependencies for MPRG monolayers are presented in Figure 4. The gaseous monolayer region will be discussed first. The cohesion pressure (Π*) values for gaseous monolayers of these surfactants were calculated according to eq 1, using the critical point parameters Πc and Ac. The value ω ) 0.22 nm2 was assumed for both surfactants. It follows from Figures 3 and 4 that these values of Π* are characteristic for the gaseous MSRG monolayer region at A ) 0.44-0.67 nm2 and the same for MPRG at A ) 0.33-0.64 nm2. The data presented in the tables show that the Π* values for these surfactants do not depend on A in the region A > 0.45 nm2, i.e., for Π < 10 mN/m. It follows then that eq 1 satisfactorily describes the experimental results, while the range of possible variations of area per molecule is rather narrow. The values of the coefficient K1 ) K* exp(-Πc∆ω/kT) calculated from the parameters of the critical points by using eq 24 for ω ) 0.22 nm2 are presented in Tables 1 and 2. It is seen that the coefficient K1 decreases with the increase of Πc. The values of the coefficient K* are, however, constant. Then, setting the values ∆ω ) 0.9 0.2 nm2 for MPRG and ∆ω ) 0.8 ( 0.2 nm2 for MSRG, one obtains K* ) 9 for both surfactants. For these values of K* and ∆ω, eq 24 agrees with the experimental data for gaseous monolayers at Π < 10 mN/m, that is, in the same range where Volmer’s equation (eq 1) is valid. However, in contrast to Volmer’s equation which cannot be used at the values of Π defined above for A > 0.68 nm2 (because it leads to negative Π values in this range), there are no restrictions for eq 24 in the range of high A values. The
Figure 4. Dependence of surface pressure on the area per molecule for 1-monopalmitoyl-rac-glycerol. Experimental results6 are presented by curves; symbols correspond to the calculations using eq 26 for ω ) ωf (Table 2).
values of K1 listed in Tables 1 and 2, together with the dependencies of K1 on A/A1/2 shown in Figure 4, allow the estimation of the aggregation number in the region of gaseous clusters, which was found to be approximately n ) 10-20. We consider next the agreement between the theoretical results and experimental data in the region of two-dimensional phase transition. The theoretical values of Π for MSRG in the region of ω < A e Ac, calculated using eq 26 for two values of ω, are shown in Figure 3. One of these values was ω ) 0.22 nm2 which corresponds to the area per molecule for an extremely compressed monolayer (the pressure is close to the collapse pressure which is approximately equal to 50 mN/m). In the second case, the value ω ) ωf was adjusted using the fitting program which minimizes the difference between the experimental values of Π and those calculated from eq 26 in the chosen range of A, usually for Ac g A g 0.26-0.28 Å2. The values of ωf obtained for MSRG are presented in the third column of Table 1. It can be seen from Figure 3 that at low temperatures the results of the calculations for ω ) 0.22 nm2 and ω ) ωf are close to each other within almost the whole
15482 J. Phys. Chem., Vol. 100, No. 38, 1996 region of two-dimensional phase transition, i.e., for both liquidexpanded and liquid-condensed films. At the same time, when the temperature increases, such an agreement can be observed in the initial stage of phase transition only. With respect to the values of ωf some notes are to be made: (1) The ωf values for MSRG are within the interval of (20% with respect to the value ω ) 0.22 nm2. (2) The decrease of ωf with the increase in the temperature (see Table 1) is consistent with the calculations of the area per molecule in the monolayer as a function of the temperature using the molecular dynamics method.24,25 The significance of this fact, however, must not be overestimated. (3) As was mentioned above, the error involved in the theoretical model used to derive eq 26 increases with the increase of Π. Therefore, the value of ωf can differ from ω ) 0.22 nm2. This difference is more significant for higher Πc and larger variation of Π during the phase transition. Similar conclusions are valid also for the MPRG monolayers, cf. Table 2 and Figure 4. For small Π values of ωf are rather close to ω ) 0.22 nm2, while for Π > 10 mN/m these values are significantly lower than 0.22 nm2. In general, eq 26 provides the correct description of the surface pressure dependence on Πc and A up to the condensed monolayer stages. In fact, this equation contains one adjustable parameter ω, which in the region of low temperatures (low Π values) is close to the value of actual area per molecule in an extremely compressed monolayer. Conclusions A new equation of state is derived which describes the twodimensional phase transition between the gaseous and condensed state of insoluble monolayers. The equilibrium between the aggregates and monomers was treated using Buttler’s equation for the chemical potential of monomers and aggregates within the surface. This allows the consideration of the effect of surface tension in the theoretical model and the estimation of the bias involved in the resulting equation of state of the monolayer. A similar approach can also be used for the gaseous monolayer. Here the differences in the monolayer behavior with respect to that of the ideal two-dimensional gas are related to the formation of small aggregates, while intermolecular interaction is not taken into account. The results predicted by the theory are in good agreement with the experimental monolayer isotherms for different temperatures. In the low-temperature region, where the assumptions used in the model are valid with a rather high degree of accuracy, the Π dependence on A in the two-dimensional phase transition region is described by the equation of state derived
Fainerman et al. here without any adjustable parameters involved. Very recently a theoretical model has been developed which describes the adsorption kinetics for self-assembling systems on the surface and in the aqueous solution.26 This theoretical model allows also the estimation of two-dimensional aggregation rate constants from the data of dynamic Π-A experiments for Langmuir monolayers of amphiphiles and considers weak Π-A dependence on the compression rate. A further improvement of the agreement between the experimental and theoretical data in a wider Π and A range up to a very tight molecular packing may most likely require the consideration of the intermolecular interactions within the monolayer states. Acknowledgment. One of the authors (D.V.) acknowledges gratefully financial assistance from the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 312, Vo 510/1-3) and the Fonds der Chemischen Industrie. References and Notes (1) Overbeck, J. A.; Ho¨nig, D.; Mo¨bius, D. Langmuir 1993, 9, 555. (2) Vollhardt, D.; Gehlert, U.; Siegel, S. Colloids Surf. A 1993, 76, 187. (3) Gehlert, U.; Weidemann, J.; Vollhardt, D. J. Colloid Interface Sci. 1995, 174, 392. (4) Sankaram, M. B.; Marsh, D.; Thompson, T. E. Biophys. J. 1992, 63, 340. (5) Zhu, J.; Eisenberg, A.; Lennox, R. B. Macromolecules 1992, 25, 6547. (6) Gutberlet, T.; Vollhardt, D. J. Colloid Interface Sci. 1995, 173, 429. (7) Miller, A.; Mo¨hwald, H. J. Chem. Phys. 1987, 86, 4258. (8) Flo¨rsheimer, M.; Mo¨hwald, H. Colloids Surf. 1991, 55, 173. (9) Ahuja, R. C.; Caruso P.-L.; Mo¨bius, D. Thin Solid Films 1994, 242, 195. (10) Barber, M. N. Phys. Rep. 1980, 59, 375. (11) Vollhardt, D. AdV. Colloid Interface Sci. 1993, 47, 1. (12) Ruckenstein, E.; Bhakta, A. Langmuir 1994, 10, 2694. (13) Israelachvili, J. N. Langmuir 1994, 10, 3774. (14) Ruckenstein, E.; Li, B. Langmuir 1995, 11, 3510. (15) Volmer, M. Z. Phys. Chem. 1925, 115, 253. (16) Gaines, G. L. J. Chem. Phys. 1978, 69, 2627. (17) Semenov, A. N. Macromolecules 1993, 26, 6617. (18) Goedel, W. A.; Wu, H.; Fridenberg, C.; Fuller, G. G.; Foster, M.; Frank, C. W. Langmuir 1994, 10, 4209. (19) Buttler, J. A. V. Proc. R. Soc. London Ser. A 1932, 138, 348. (20) Lucassen-Reynders, E. H. J. Phys. Chem. 1966, 70, 1777. (21) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1972, 41, 156; 1982, 85, 178. (22) Fainerman, V. B.; Miller, R.; Wu¨stneck, R.; Makievski, A. V. J. Phys. Chem., in press. (23) Retter, U. J. Electroanal. Chem. 1993, 349, 49. (24) Karaborni, S.; Tenside, Surfactants Deterg. 1993, 30, 256. (25) Karaborni, S; Verbist, G. Europhys. Lett. 1994, 27, 467. (26) Fainerman, V. B.; Vollhardt, D.; Melzer, V. Submitted for publication in J. Chem. Phys.
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