Langmuir 2006, 22, 1701-1705
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Surface-Pressure Isotherms of Monolayers Formed by Microsize and Nanosize Particles V. B. Fainerman,† V. I. Kovalchuk,‡ E. H. Lucassen-Reynders,§ D. O. Grigoriev,| J. K. Ferri,⊥ M. E. Leser,# M. Michel,# R. Miller,*,| and H. Mo¨hwald| Medical Physicochemical Centre, Donetsk Medical UniVersity, 16 Ilych AVenue, 83003 Donetsk, Ukraine, Institute of Biocolloid Chemistry, 42 Vernadsky AVenue, 03680 KyiV (KieV), Ukraine, Mathenesselaan 11, 2343 HA Oegstgeest, The Netherlands, Max-Planck-Institut fu¨r Kolloid- und Grenzfla¨chenforschung, 14424 Potsdam/Golm, Germany, Department of Chemical Engineering, Lafayette College, Easton, PennsylVania 18042, and Nestec Ltd., Nestle´ Research Centre, Vers-chez-les-Blanc, CH-1000 Lausanne 26, Switzerland ReceiVed September 3, 2005. In Final Form: NoVember 22, 2005 The thermodynamic model of a 2D solution developed earlier for protein monolayers at liquid interfaces is generalized for monolayers composed of micro- and nanoparticles. Surface pressure isotherms of particle monolayers published in the literature for a wide range of particles sizes (between 75 µm and 7.5 nm) are described by the theoretical model with one modification. The calculations of surface pressure Π on area A provide satisfactory agreement with the experimental data. The theory also yields reasonable cross-sectional area values of the solvent molecule water in the range between 0.12 and 0.18 nm2, which is almost independent of particle size. Also, the area per particle in a closely packed monolayer obtained from the theory is quite realistic.
Introduction Foams and emulsions can be formed and stabilized in the presence of well-selected surfactants, polymers, proteins, and their mixtures. A new alternative is the use of well-defined nanoparticles, mainly for the stabilization of emulsions. The idea of this principle is not really new because in 1907 Pickering used particles to stabilize emulsions.1 Now, particles in a broad range of sizes and of high quality with respect to size homogeneity and surface properties are available. Such nanoparticles can have different hydrophobicity/hydrophilicity that may change in the presence of surface-active molecules. The stability of emulsions and foams depends essentially on the viscoelasticity of the surface layer. A theoretical analysis of the dynamic dilation characteristics of a liquid interface, at which dispersed solid particles or liquid drops are present, was made by Lucassen.2,3 Under certain conditions (if no appreciable desorption of particles takes place and the wetting angle remains almost constant during the compression or expansion of the monolayer), the elasticity is determined by the shape of the dependence of surface pressure (Π) on area per particle (A) at constant temperature as E0 ) -dΠ/d ln A. Recently, the number of publications concerning experimental studies of Π-A isotherms for micro- and nanoparticle monolayers at water/air or water/oil interfaces has significantly increased.4-13
The results of studies on the behavior of particles at the liquid/ liquid interface were reviewed by Binks.14 As far as we are aware, theoretical studies of monolayers formed by nanoparticles so far have disregarded the substantial difference between particles and ordinary surfactant molecules. This difference was not taken into account in published theoretical attempts to describe nanoparticles at liquid interfaces. For example, in ref 14 monolayers of nanoparticles were analyzed using the well-known Volmer and van der Waals equations, assuming an area per molecule in the monolayer identical to the area of a nanoparticle. This approach leads to unrealistic dependencies of surface pressure on particle size. In particular, it follows from these equations that only particles smaller than 1 nm are able to create a surface pressure of about 10 mN/m at 50-70% monolayer coverage. This conclusion contradicts the experimental data obtained for micro- and nanoparticles. For the theoretical analysis of monolayers of nanoparticles, an equation is needed that accounts for the significant difference between the sizes of adsorbing particles and the solvent molecules. Such expressions for the surface tension of mixed solutions of molecules with different sizes were presented in the monograph15 and also derived recently for protein solutions.16 Equations of this type will be obtained here by a modification that allows us to describe Π-A isotherms of micro- and nanoparticles of different sizes as reported in the literature.
†
Donetsk Medical University. Institute of Biocolloid Chemistry. Mathenesselaan 11. | Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung. ⊥ Lafayette College. # Nestec Ltd. ‡ §
(1) Pickering, S. U. J. Chem. Soc. 1907, 91, 2001 (2) Lucassen, J. Colloids Surf. 1992, 65, 131. (3) Lucassen, J. Colloids Surf. 1992, 65, 139 (4) Mate, M.; Fendler, J. H.; Ramsden, J. J.; Szalma, J.; Ho´rvo¨lgyi, Z. Langmuir 1998, 14, 6501. (5) Ho´rvo¨lgyi, Z.; Ma´te´, M.; Daniel, A.; Szalma, J. Colloids Surf., A 1999, 156, 501. (6) Ho´rvo¨lgyi, Z.; Nemeth, S.; Fendler, J. H. Langmuir 1996, 12, 997. (7) Zhang, K.; Tang, F.; Jiang, L. Langmuir 1991, 7, 1293.
(8) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969. (9) Aveyard, R.; Clint, J. H.; Nees, D.; Quirke, N. Langmuir 2000, 16, 8820. (10) Lefebure, S.; Menager, C.; Cabuil, V.; Assenheimer, M.; Gallet, F.; Flament, C. J. Phys. Chem. B 1998, 102, 2733. (11) Schwartz, H.; Harel, Y.; Efrima, S. Langmuir 2001, 17, 3884. (12) Wolert, E.; Setz, S. M.; Underhill, R. S.; Duran, R. S.; Schappacher, M.; Deffieux, A.; Ho¨lderle, M.; Mu¨lhaupt, R. Langmuir 2001, 17, 5671. (13) Mayya, K. M.; Cole, A.; Jain, N.; Phadtare, S.; Langevin, D.; Sastry, M. Langmuir 2003, 19, 9147. (14) Binks, B. P. Curr. Opin. Colloid Interface Sci. 2002, 7, 21. (15) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans: London, 1966. (16) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. AdV. Colloid Interface Sci. 2003, 106, 237.
10.1021/la052407t CCC: $33.50 © 2006 American Chemical Society Published on Web 01/14/2006
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Theory In ref 16, an equation of state for surface layers was derived on the basis of 2D solution theory (in particular, on Butler’s equation17) to describe the mixture of any number of components, each characterized by its own geometry
Π)-
kT (ln x0 + ln f0) ω0
(1)
where Π is the surface pressure, k is the Boltzmann constant, T is the temperature, ω0 is the molecular area of a solvent molecule, fi represents the activity coefficients, xi ) Ni/∑ Ni represents the molar surface fractions, and Ni represents the numbers of moles of the ith component. The subscript 0 refers to the solvent. An important feature of eq 1 is that it involves the only the characteristics of the solvent because for its derivation the chemical potential of the solvent in the surface layer is set equal to that in the solution bulk, for solutions so dilute as to have a solvent molar fraction very close to 1.16 The ω0 value depends on the choice of the position of the dividing surface; eq 1 requires positive values of ω0 and N0. In ref 16, an equation defining a dividing surface was proposed that relates the surface excesses Γi of the solvent (subscript i ) 0) and dissolved and insoluble species (i g 1) with any molecular area ωi n
∑ i)0
Γi )
1 Nω0
( )∑
+ 1-
ω
n
ω0
i)1
Γi
(2)
where N is Avogadro’s number. The average molecular area of dissolved and insoluble species ω is determined from the expression
ω0 * ω )
ω1Γ1 + ω2Γ2 + ... Γ1 + Γ2 + ...
∑ ig0
(4) (θi/ni)
with ni ) ωi/ω0. The total value of fi can now be obtained from the additivity of enthalpy (H) and entropy (E) contributions:16
fi ) f Hi f Ei
or
ln fi ) ln f Hi + ln f Ei
(5)
Approximations for the enthalpic contribution to the activity coefficients were given in ref 16 in the following way
ln f H0 ) aθ2
ln f Hj ) anjθ02
(6)
where a is the Frumkin interaction parameter for the nonideality. Expressions for the entropic contribution to the activity coefficients were derived from a first-order, Flory-type model for mixtures of any number of different-area molecules in ref 18: (17) Butler, J. A. V. Proc. R. Soc., Ser. A 1932, 138, 348. (18) Lucassen-Reynders, E. H. Colloids Surf., A 1994, 91, 79.
∑ ig0 n
θi
+ ln nj
ig0
i
ni
(7)
By introducing relations 4-7 into the original expression (eq 1), an equation of state was obtained in ref 16 that describes 2D surface layers containing the solvent and any other component (e.g., protein or solid particles) and is valid for n ) ω/ω0 > 1:
-
(
)
ω0 Πω0 + aθ2 ) ln(1 - θ) + θ 1 kT ω
(8)
Here θ is the fraction of the surface covered by molecules or solid particles with an average area per entity ω. Comparing eq 8 with the expression (eq 13.24) derived in ref 15 for the surface tension in the mixture of molecules of different sizes and assuming the fraction of the dissolved component in the bulk phase to be almost zero, one can see that these equations are identical. Note that eq 13.24 was derived in ref 15 using an approach entirely different from that employed to derive eq 8 in ref 16: a statistical mechanics formulation was applied in ref 15 involving the calculation of the configuration energy of a mixture, the free energy, and the chemical potentials of the components at the surface, from which the surface tension of a mixed monolayer was then obtained in the framework of the so-called parallel model. Note that despite the differences in the theoretical methods in refs 15 and 16, the physical ideas of the derived equations of state are quite similar. Introducing the available surface area per micro- or nanoparticle A via the relation θ ) ω/A and assuming that the enthalpy contribution is independent of θ (i.e., taking ln f H0 to be constant, which corresponds to a liquid-expanded monolayer19), under the condition n ) ω/ω0 . 1 one obtains from eq 8 an expression for the Π-A isotherm for a particle monolayer:
Π)-
θj nj
( ) [ ∑( )] θi
(3)
Equation 1 can be transformed into a more convenient form if one expresses the molar fraction of the surface layer components via the surface fractions θj occupied by the jth component, as follows from eq 2
xj )
ln f Ej ) 1 - nj
ω ω kT ln 1 + - Πcoh ω0 A A
[(
) ( )]
(9)
Here Πcoh ) kT/ω0 ln f H0 is the cohesion pressure. Note that in agreement with the theory of solutions for binary bulk systems or monolayers, the activity coefficient includes solvent-solvent, solvent-solute, and solute-solute interactions.20 Although we do not specifically analyze the impact of Coulomb interactions on the surface pressure, its contribution to Πcoh is included. Thus, Πcoh includes all characteristics of long-range forces between all components.21 Let us draw attention to a peculiarity of eq 9, which is an analogue to all other equations of state of insoluble monolayers.19-21 Equation 9 yields positive values for Π and is therefore applicable only at surface coverages above a minimum value of (ω/A)min, determined by the condition Π ) 0. For (ω/A) < (ω/A)min (i.e., A > Amin), the contributions of ideal and nonideal entropy are compensated for by the interaction between the components (Πcoh). As one can see from eq 9, the surface pressure does not depend on the particle size but is determined by the monolayer coverage ω/A and the parameters ω0 and Πcoh. Because eq 9 involves the ratio ω/A, the corresponding areas can refer not only to the individual particles but also to the areas occupied by a certain mass of particles or, for example, to the entire monolayer in a Langmuir trough. (19) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 1999, 103, 145. (20) Read, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. (21) Rusanov, A. I. J. Chem. Phys. 2004, 120, 10736.
Surface-Pressure Isotherms of Monolayers
Langmuir, Vol. 22, No. 4, 2006 1703
Equation 9 can also be derived in another way. If we use the chemical potential µ1 of amphiphilic molecules or particles (both soluble and insoluble ones) in the surface layer instead of the expressions for the chemical potentials of the solvent in the surface layer and the bulk phase and the Gibbs’ adsorption equation dΠ ) Γ1dµ1 as an additional relationship, then we obtain19
dΠ )
RT θ d(ln x1 + ln f1 - ln f10) ω 1-θ
(10)
Here the activity coefficient f10 refers to the standard state at infinite dilution. Moreover, the molar fraction of particles within the surface x1 can be expressed via the monolayer coverage θ. From eq 4, it follows that
x1 )
θ n1 - n1θ + θ
(11)
Equation 7 yields the expressions for the entropy contribution to the activity coefficient:
ln f E1 ) ln[n1 + (1 - n1)θ1] + (1 - n1)(1 - θ1) (12) ln f E10 ) ln n1 + (1 - n1)
(13)
Introducing expressions 11-13 into eq 10 results in an equation that is rather complicated to integrate. This equation, however, can be simplified for the case of n1 ) ω/ω0 . 1. The integration of the simplified eq 10 (assuming that ln f H0 ) const, see ref 19) results in the equation of state:
Figure 1. Dependence of surface pressure Π on the Langmuir trough area A for silanized glass particles of 75 ( 5 µm diameter with different degrees of hydrophobization (A, B, C, D, and E, thin lines); data according to ref 5; theoretical curves calculated from eq 9 (-) and eq 15 (- -) with model parameters listed in Table 1. Table 1. Parameters of Theoretical Models for Silanized Glass Particles 75 µm in Diameter According to the Data5,a parameters
A
B
C
D
E
ω, cm2 ω0, nm2 Πcoh, mN/m contact angle φ, °
50/57 0.4/0.16 19.7/17.0 39
54/61 0.42/0.17 16.5/13.0 62
54/61 0.43/0.17 13.6/9.5 71
60/67 0.44/0.18 12.4/8.0 76
60/68 0.44/0.18 6.9/3.5 87
ω ω kT ω kT ln 1 ln 1 + - Πcoh (14) ω A ω0 A A
a Values shown as numerator/denominator correspond to eq 15/eq 9, respectively.
Because ω . ω0, the first term on the right-hand side can be neglected, and eq 14 becomes identical to eq 9, which was derived by the other method. For Π-A isotherms of insoluble (Langmuir) monolayers of amphiphilic molecules, assuming association or dissociation of these molecules in the surface layer, a generalized Volmer equation was derived (on the basis of eq 10) in refs 19, 22, and 23 and has the form
to the entire monolayer in the Langmuir trough, and different experimental curves correspond to particles of different degrees of hydrophobicity. The theoretical isotherms calculated from eqs 9 and 15 by best fit are also shown in Figure 1, and the parameters of the equations are summarized in Table 1, along with the corresponding wetting angles φ as given in ref 5. It is seen that the two theoretical models agree well with the experimental data for all studied microparticles. However, considering the values of the parameters listed in Table 1 one can see that eq 9 gives more realistic values. For example, the ω values (for the entire monolayer) calculated from eq 15 are 10-15% lower than the experimental ones, whereas those calculated from eq 9 correspond exactly to the values obtained in ref 5. The ω0 values obtained for the two models have the correct order of magnitude; however, the values calculated from eq 9 are closer to the cross-sectional area of a water molecule in the bulk phase (ca. 0.1 nm2). Note that when the position of the dividing surface is chosen according to eq 2, the area per solvent molecule can differ somewhat from the cross-sectional area. Figure 2 shows the Πcoh dependencies on cos φ calculated using eqs 9 and 15. There exists an almost linear interrelation between the nonideality of enthalpy, given by the parameter Πcoh, and the wetting angle (cos φ). Therefore, the more hydrophilic the particles, the higher the Πcoh. In ref 24, it is shown that the attraction between particles increases with their hydrophobicity because of hydrophobic interaction. These results show the complex character of Πcoh, which is regarded not only for the interaction between the particles but also for all other interactions in the surface layer. Strong attractions between particles can be the reason for the transition from liquid-expanded
Π)-
Π)
(
)
[(
) ( )]
kT(ω/A) nkT - Πcoh ) - Πcoh A-ω ω0(1 - ω/A)
(15)
It is seen that eq 15, which is an approximation of eq 9 assuming low monolayer coverage and neglecting the nonideality of entropy, is also able to describe the behavior of monolayers composed of particles of any size. Similar to eq 9, this equation involves not the geometric parameters of amphiphilic molecules (or particles) but only the coverage of the monolayer by these entities. Equation 15 provides a good description of experimental results obtained for various systems. For example, for some insoluble proteins in a liquid-expanded monolayer state, a value of n ) 20-100 was obtained in ref 23. Let us consider now some sets of experimental data from the literature.
Results and Discussion The experimental Π-A isotherms obtained for silanized glass particles (75 ( 5 µm diameter) in a broad hydrophobicity range (40-90°)5 are presented in Figure 1. In this Figure, area A refers (22) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 2003, 107, 3098. (23) Wu¨stneck, R.; Fainerman, V. B.; Wu¨stneck, N.; Pison, U. J. Phys. Chem. B 2004, 108, 1766.
(24) Ho´rvo¨lgyi, Z.; Ma´te´, M.; Zrı´nyi, M. Colloids Surf., A 1994, 84, 207.
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Fainerman et al.
Figure 2. Dependence of Πcoh on cos φ for silanized glass particles of 75 ( 5 µm diameter for Πcoh values calculated from eq 9 (9) and eq 15 (4).
Figure 4. Dependence of surface pressure Π on the area per particle A in the monolayer for monodisperse magnetic particles of 7.5 nm (A) and 11 nm (B) diameter; data according to ref 10; (- -) first compression of the monolayer; (-) third (A, diameter 7.5 nm) or fourth (B, diameter 11 nm) monolayer compression cycle; theoretical curves ()) calculated from eq 9 with model parameters listed in Table 3. Table 3. Parameters of the Theoretical Model (Eq 9) for Monodisperse Magnetic Particles 7.5 and 11 nm in Diameter According to the Data10
Figure 3. Dependence of surface pressure Π on the monolayer coverage for polymeric particles 113 nm in diameter without dispersant (4) and with dispersant (9); data according to ref 12; theoretical curves calculated from eq 9 with model parameters listed in Table 2. Table 2. Parameters of the Theoretical Model (Eq 9) for Polymer Particles 113 nm in Diameter According to the Data12 model parameters
without dispersant
with dispersant
ω, m2/g ω0, nm2 Πcoh, mN/m
15.5 0.12 7.0
23 0.12 10.0
to liquid-condensed or solid monolayer behavior.25 Such monolayer states are not described by eq 9. Let us consider now the Π-A isotherms (here A is the area per unit particle mass) for polymeric spherical particles of 113 nm diameter.12 The results obtained for the monolayer of these particles with and without a copolymer as dispersant are shown in Figure 3. The model parameters for the two types of monolayers are listed in Table 2. The theoretical curves presented in Figure 3 were calculated as the best fit from eq 9. The calculations according to eq 15 result in almost the same curves (similar to Figure 1), but the corresponding values of ω0 are somewhat higher than those calculated from eq 9 and the values of ω are somewhat lower than those obtained from eq 9. Note that for both the microparticles of 75 µm diameter (cf. Figure 1) and for the polymeric particles of 113 nm size the ω0 values obtained (25) Tolnai, G.; Agod, A.; Kabai-Faix, M.; Kova´cs, A. L.; Ramsden, J. J.; Ho´rvo¨lgyi, Z. J. Phys. Chem. B 2003, 107, 11109.
model parameters
D ) 7.5 nm
D ) 11 nm
ω, nm2 ω0, nm2 Πcoh, mN/m
47.5 0.13 1.5
104 0.12 0.5
from the theory are almost the same and are almost equal, provided the reservations following from eq 2 are maintained, to the crosssectional area of a water molecule. It is seen from Figure 3 that for the individual monolayer of polymeric particles the theory well describes the experimental curve over a very wide range of surface pressure up to the monolayer collapse. At the same time, for the mixture of polymeric particles with the dispersant, the agreement with the experiment is seen to be good only at low surface pressure values. Presumably, for such systems more complicated theoretical models that assume the presence of a second soluble component in the mixture should be used. Such models were developed recently for protein/surfactant mixtures in refs 26 and 27. Figure 4 illustrates data obtained in ref 10 for monodisperse magnetic particles of 7.5 and 11 nm diameter, where the values of A were calculated per particle. The monolayers of magnetic particles studied in ref 10 contained lauric acid. Therefore, during the first monolayer compression cycle (dashed line) the surface pressure was higher than for subsequent cycles because of the dissolution of the acid into the subphase during the monolayer compression/expansion cycles. However, from the second compression/expansion cycle on, the difference between the Π-A isotherms becomes insignificant. The compression isotherms for the third (particle diameter 7.5 nm) and fourth (particle diameter 11 nm) cycles of the monolayer compression/expansion are shown in Figure 4. The theoretical curves for these isotherms were calculated from eq 9 using the parameters listed in Table 3. The resulting agreement between theory and experiment is quite satisfactory. One possible reason for the slight difference is the presence of small amounts of lauric acid in the monolayer due (26) Fainerman, V. B.; Zholob, S. A.; Leser, M. E.; Michel, M.; Miller, R. J. Colloid Interface Sci. 2004, 274, 496. (27) Fainerman, V. B.; Zholob, S. A.; Leser, M. E.; Michel, M.; Miller, R. J. Phys. Chem. B 2004, 108, 16780.
Surface-Pressure Isotherms of Monolayers
to the incomplete desorption into the subphase. The ω values calculated from eq 7 are almost equal to those reported in ref 10 for the area per particle: 50 and 105 nm2 for particles 7.5 and 11 nm in diameter, respectively. It is also seen from Table 3 that the calculated molecular area ω0 for the solvent is quite realistic and does not depend on the particle size. Equation 9 also provides a satisfactory description of the Π-A isotherm for monodisperse spherical polystyrene particles of 2.6 µm diameter at the water/octane interface.8,9 For this system, the fitting according to eq 9 yields ω0 ) 0.12 nm2, ω ) 32 cm2 (calculated for the entire Langmuir trough area), and a Πcoh value close to zero.
Conclusions The thermodynamic model of 2D solutions developed earlier16 for monolayers of protein molecules at liquid interfaces is
Langmuir, Vol. 22, No. 4, 2006 1705
generalized for monolayers composed of micro- and nanoparticles. The theoretical isotherm expressed by eq 9 is used to analyze the results reported in the literature as surface-pressure isotherms of monolayers for a wide range of particle sizes between 75 µm and 7.5 nm. The calculations not only provide satisfactory agreement with experimental Π-A isotherms but also produce reasonable area values for the solvent molecule in the range of 0.12 to 0.18 nm2, which is correct and almost independent of the particle size. Also, the area per particle in a closely packed monolayer that follows from eq 9 is very realistic. Acknowledgment. This work was financially supported by projects of the European Space Agency (FASES MAP AO-99052), the DFG (Mi418/14), and the Ukrainian SFFR (project no. 03.07/00227). LA052407T
