Langmuir 2001, 17, 3103-3108
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Molecular Dynamics Simulation of the Surface Pressure of Colloidal Monolayers Jizhong Sun and T. Stirner* Department of Chemistry, University of Hull, Hull HU6 7RX, United Kingdom Received November 13, 2000. In Final Form: March 1, 2001
The compression of two-dimensional colloidal monolayers consisting of polystyrene particles at an oilwater interface is investigated theoretically. Expressions for the pair-particle interactions are derived both for point particles and for particles of radius R. The resulting expressions for the interaction energies are employed in a molecular dynamics simulation of the surface pressure of these monolayers. A comparison of the theoretical results with recent observations allows the determination of the dominant interaction mechanisms in the experimental system. Possible physical interpretations of the interaction mechanisms between the particles are discussed.
1. Introduction Recently the behavior of particle monolayers at liquid surfaces has drawn wide interest, arising from the ability of the particles to modify the stability of emulsions, foams, and interfacial properties.1-3 For instance, Pieranski4 reported a system of polystyrene particles, trapped at the interface between an aqueous phase and air, forming a single monolayer in a triangular lattice, and suggested that dipole-dipole interactions are responsible for the monolayer formation. Robinson and Earnshaw5 investigated a similar system and proposed the existence of dipoles at the particle-air interface to explain the stability of the monolayers against aggregation even at high electrolyte concentrations. Very recent experiments by Aveyard et al.,6 however, on monolayers of polystyrene particles at air-water and oil-water interfaces exhibit many phenomena which are difficult to explain by dipoledipole interactions only. In short, Aveyard et al.6 studied the compression and structure of sulfate polystyrene latex particle monolayers at air-water and octane-water interfaces. They found that the particle monolayers formed a fairly ordered structure (resulting from the interparticle repulsions) which, on compression in a Langmuir trough, underwent a phase transition from a hexagonal to a rhombohedral lattice. They also found that, with increasing electrolyte concentration, the particles at the airwater interface formed 2D clusters, while the particle monolayers at the oil-water interface remained highly ordered.6 Theoretically, Kalia and Vashishta7 and Zangi and Rice8 studied the melting transition of a colloidal monolayer using dipole-dipole interactions and a Marcus-Rice potential, respectively, with molecular dynamics (MD) simulations. Similarly, Terao and Nakayama9 investigated (1) Aveyard, R.; Clint, J. H. J. Chem. Soc., Faraday Trans. 1995, 91, 2681. See also J. Chem. Soc., Faraday Trans. 1996, 92, 85. (2) Aveyard, R.; Cooper, P.; Fletcher, P. D. I.; Rutherford, C. E. Langmuir 1993, 9, 604. (3) Aveyard, R.; Clint, J. H.; Nees, D. Colloid Polym. Sci. 2000, 278, 155. (4) Pieranski, P. Phys. Rev. Lett. 1980, 45, 569. (5) Robinson, D. J.; Earnshaw, J. C. Langmuir 1993, 9, 1436. (6) Aveyard, R.; Clint, J. H.; Nees, D.; Paunov, V. N. Langmuir 2000, 16, 1969. (7) Kalia, R. K.; Vashishta, P. J. Phys. C: Solid State Phys. 1981, 14, L643. (8) Zangi, R.; Rice, S. A. Phys. Rev. E 1998, 58, 7529.
the crystallization in two-dimensional colloidal systems with Monte Carlo simulations. Hurd10 investigated the interactions between charged particles at the air-water interface theoretically. Also, Aveyard et al. derived an analytic expression for the electrostatic surface pressure of colloidal monolayers of hexagonal structure6 and used a harmonic force model to describe the monolayer collapse.11 Finally, Bresme and Quirke12-14 studied the wetting behavior and line tension of nanometer-size particulates at a liquid-vapor interface, and Fenwick, Bresme, and Quirke15 simulated Langmuir trough experiments on nanoparticulate arrays using MD simulations and thermodynamic arguments. They found that, at the onset of surface collapse, the monolayer of particulates buckles under compression.15 The aim of the present study is to establish a method by which the particle interactions in colloidal monolayers can be investigated theoretically. For this purpose, the expressions for the interaction energies between the colloidal particles are derived. In the present work we will focus our attention on the situation where the electrostatic interactions, that is, charge-charge, chargedipole, and dipole-dipole interactions, dominate and where the particle separations are sufficiently large to neglect the van der Waals interactions (that is we will not consider particle aggregation). To date most theoretical investigations assumed point particles when dealing with particle interaction energies. However, this assumption may be invalid for large colloidal particles and/or small interparticle separations. Therefore, in the present work, the interaction energies are derived both for point particles and for particles of radius R. The resulting energy expressions are then employed in computer simulations utilizing the MD technique. In particular, the model is used to calculate the surface pressure of colloidal monolayers consisting of micrometer-size particles. Finally, the results of the calculations are compared with the recent experimental observations of Aveyard et al.6 (9) Terao, T.; Nakayama, T. Phys. Rev. E 1999, 60, 7157. (10) Hurd, A. J. J. Phys. A: Math. Gen. 1985, 18, L1055. (11) Aveyard, R.; Clint, J. H.; Nees, D.; Quirke, N. Langmuir 2000, 16, 8820. (12) Bresme, F.; Quirke, N. Phys. Rev. Lett. 1998, 80, 3791. (13) Bresme, F.; Quirke, N. Phys. Chem. Chem. Phys. 1999, 1, 2149. (14) Bresme, F.; Quirke, N. J. Chem. Phys. 1999, 110, 3536. (15) Fenwick, N.; Bresme, F.; Quirke, N. J. Chem. Phys. (in press).
10.1021/la001574k CCC: $20.00 © 2001 American Chemical Society Published on Web 04/14/2001
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Sun and Stirner
Ecc )
4πR4C2c r
(2)
for the charge-charge interaction energy between the two particles. It is interesting to note that this expression is identical to the interaction energy between two point charges with the same total charge.16 The interaction potential (per two unit surface areas) between two point dipoles, P1 and P2, separated by a distance r can be written as17 Figure 1. Schematic diagram of the geometry used to describe the electrostatic interaction between an area element Q on the surface of a sphere of radius R and a second area element located at P.
2. Theory Since we are ultimately interested in modeling the experimental system described by Aveyard et al.,6 we will be able to justify some of the assumptions made in the derivation of the particle interaction energies directly from the experimental conditions. However, some approximations have to be justified and/or discussed a posteriori. We will return to this point in section 5. From the experiments6 we know that the colloidal particles are trapped in a deep surface potential well. We can therefore ignore the motion of the colloidal particles perpendicular to the surface as long as the particle density is not too high. Furthermore, in the system studied here the colloidal particles are relatively large (i.e. having a radius of R ) 1.3 µm), and we can thus neglect the influence of Brownian motion on the surface pressure of the monolayers. We now turn to the derivation of the pair-potential energies for the charge-charge, dipole-dipole, and charge-dipole interactions. Consider two perfect spheres of radius R (whose centers are located at points O and P) in an isotropic dielectric medium of permittivity separated by a center-to-center distance OP ) r (see Figure 1). The total interaction energy between the two particles can be written as
E)
∫S ∫S V ds1 ds2 1
2
(1)
where ds1, ds2 and S1, S2 denote area elements and total surface areas of the two particles, respectively, and V is the interaction potential between two unit surface areas. The area element Q of a ring on the surface of one of the particles, described by a cone of angle POQ ) Θ and axis of rotation OP, is R2 sin Θ dΘ dφ, where φ is the azimuthal angle. Integrating this area element from φ ) 0 to 2π gives the area of the ring as 2πR2 sin Θ dΘ. Hence, we can write the interaction energy between one sphere and a unit area located at P as
C2c 4π
2
∫0π 2πR
sin Θ dΘ r′
where r′2 ) R2 + r2 - 2Rr cos Θ is the distance between P and Q, where the interaction potential (per two unit surface areas) is
Vcc )
C2c 4πr′
and where we have assumed that each sphere carries a charge which is uniformly distributed over the surface of the sphere with charge density Cc. Integrating over the surface areas of both spheres gives the result
V12 )
P1‚P2 - 3(n‚P1)(n‚P2) 4πr3(4πR2)2
(3)
where n is a unit vector in the direction (r1 - r2) and r1(r2) describes the location of particle 1 (particle 2). Here we assume that the dipoles are parallel (i.e. P1||P2), of equal magnitude (i.e. |P1| ) |P2| ) P), and perpendicular to n (i.e. P ⊥ n). Hence eq 3 becomes
V12 )
P2 4πr (4πR2)2 3
(4)
Utilizing this relationship for the interaction potential and carrying out the same integration over the surface areas of the two spheres as described above gives
Edd )
P2 [2 ln r - ln(r + 2R) - ln(r - 2R)] (5) 16πrR2
for the dipole-dipole interaction energy in this case (i.e. where dipoles are considered to be uniformly distributed over the particle surface and normal to the surface). The interaction potential (per two unit surface areas) between a charge and a dipole is
Vcd )
CcP 4πr2(4πR2)
(assuming again that the dipole moment vector is perpendicular to n). Using this dependence in the integration procedure described above gives
Ecd )
CcP [(r + 2R) ln(r + 2R) + 4r (r - 2R) ln(r - 2R) - 2r ln r] (6)
for the interaction energy between a particle with surface charge density Cc and a dipole of moment P. We note that, in the derivation of the interaction energies, we clearly oversimplify the situation by assuming that the particles are located in one dielectric medium only. However, we expect (and demonstrate below) that the derived expressions will nevertheless give a valid description of the interparticle interactions, if we assume that the surface charge density Cc and the magnitude of the dipole moment P are two fitting parameters in the calculations. We will also come back to this point in section 5. To illustrate the difference between the interaction energies assuming point particles and particles of radius R, we have evaluated the corresponding energy ratios for the two cases. The ratios for the various types of interac(16) Plonsey, R.; Collin, R. E. Principles and applications of electromagnetic fields; McGraw-Hill: Tokyo, 1961; p 49. (17) Jackson, J. D. Classical electrodynamics; Wiley: New York, 1975; p 143.
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Figure 2. Relative interaction energies (for point particles and particles of radius R as described in the text) for chargecharge (dashed line), charge-dipole (dotted line), and dipoledipole (full line) interactions as a function of relative interparticle separation (β ) r/2R).
tions, that is, charge-charge, charge-dipole, and dipoledipole, are given by
Ecc Epcc Ecd Epcd
(
) β ln
)1
(7)
)
(1 + β-1)1+β
(8)
(1 - β-1)1-β
and
Edd Epdd
(
) β2 ln
β2 (1 + β)(1 - β)
)
(9)
respectively, where the superscript p denotes the interaction energy between two point particles and β ) r/2R. The corresponding energy ratio versus β curves for these equations are displayed in Figure 2. The dashed line in Figure 2 represents the ratio for the charge-charge interaction, the dotted line the charge-dipole interaction, and the full line the dipole-dipole interaction. As was noted above, the expressions for the charge-charge interaction for point particles and for particles of radius R are identical. For the cases of dipole-dipole and chargedipole interactions, it can be seen from Figure 2 that if the separation between the two dipoles or the charge and the dipole, respectively, is much larger than the particle size, then the energy expressions applicable to particles of radius R approach those for point particles. However, for relatively small particle separations (i.e. less than approximately three particle diameters) the interaction energies for particles of radius R, in particular for the dipole-dipole interaction, are significantly enhanced over those for point particles. 3. Computer Simulation Details To simulate a dynamic process, such as the compression of the colloidal monolayers at the oil-water interface, realistically (that is, to give the colloidal particles enough time to relax into a local equilibrium configuration during the movement of the Langmuir trough wall), we implemented the theoretical model described above using the MD technique on a computer. The pair-interaction energies in eqs 2, 5, and 6 were employed in the MD simulations; that is, only electrostatic interactions were taken into account. The corresponding forces can be obtained by differentiation of the interaction energies with respect to the particle separation r. The resulting evolution of this many-particle system can now be described by
Figure 3. Convergence of calculated surface pressure Π (in % of Π at N ) 40 000) as a function of the number of particles N in the simulation cell (R ) 1.3 µm, dipole-dipole interactions).
Newton’s equations of motion which were integrated numerically using the velocity Verlet algorithm.18 Here only the interactions between the particles through the oil phase were considered, since the relative permittivity of water is 78 at room temperature, compared with a relative permittivity of oil of 2. In this respect it is also important to note that the experiments were carried out for various electrolyte concentrations in the aqueous phase, where it was observed that NaCl concentrations which led to considerable particle aggregation at the air-water interface (due to electrostatic screening effects) had a negligible effect on the surface pressure-surface area isotherms of the particle monolayers at the oil-water interface.19 To make the water-oil-particle-monolayer system tractable on a computer, only the interactions between the colloidal particles were evaluated in the MD simulations, while the oil phase was simulated by its static dielectric constant in the particle interaction energies. Calculating the pressure can be achieved directly via the virial theorem.20 For the present two-dimensional system, we can define the surface pressure, Π, relative to the oil-water interface without colloidal particles as
ΠA ) NkT +
〈 ∑∑ 〉 1
N
N
2 i)1 j>1
rij ‚fij
(10)
where k is the Boltzmann constant, T is the temperature, and fij (rij) is the force (separation) between particles i and j. The MD simulations of the monolayer compression were performed for N ) 1000 colloid particles in a rectangular cell with periodic boundary conditions. The minimum image convention18 without a potential cutoff radius was employed in the calculations. Initially, the cell area, A, was determined by the particle coverage (i.e. A ) πR2N/ coverage). The monolayer compression was simulated by incrementing the coverage in steps of 0.05, which took typically 10 000 time steps, with a time step of 40 ns. The system was then equilibrated for ≈5000 time steps at each value of the coverage. Convergence of the simulation results was checked by varying the number of particles N in the cell. An example convergence test of Π for coverages of 0.3 (open circles) and 0.5 (open squares) is shown in Figure 3. As can be seen from this figure, the calculated values of Π for N ) 1000 are ≈90% converged, which we considered to be sufficient in view of the large (18) Allen, M. P.; Tildesley, D. J. Computer simulation of liquids; Clarendon Press: Oxford, 1992. (19) Aveyard, R.; Clint, J. H.; Nees, D. Colloid Polym. Sci. 2000, 278, 155. (20) Rowlinson, J. S.; Widom, B. Molecular theory of capillarity; Clarendon Press: Oxford, 1982; Chapter 4.3.
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Figure 4. Calculated surface pressure Π (normalized to Π for point particles at a coverage of 0.1) for a monolayer of point particles (dashed line) and R ) 1.3 µm-size particles (full line) under dipole-dipole interactions as a function of coverage.
Figure 5. Example configuration of the colloidal monolayer at a coverage of 0.35 (R ) 1.3 µm, dipole-dipole interactions).
Sun and Stirner
Figure 7. Calculated surface pressure for a particle monolayer (1.3 µm particle radius) under charge-charge (dashed line), charge-dipole (dotted line), and dipole-dipole (full line) interactions as a function of coverage (normalized as described in the text). The experimental data points (full circles) are taken from ref 6.
identical. However, as can be seen from Figure 4, assuming point particles in the derivation of the interaction energies can lead to a considerable underestimation of the surface pressure values at high coverages. We also note that the MD simulations reproduced successfully the observed hexagonal close-packed (HCP) lattice structure.6 An example configuration of the colloidal monolayer at a coverage of 0.35 is shown in Figure 5. The corresponding pair correlation function g(r), evaluated from the expression18
g(r) )
Figure 6. Pair correlation function g(r) corresponding to the example configuration in Figure 5 as a function of particle separation r.
increase in computational effort for large particle systems (note the logarithmic scale for N). 4. Results and Discussion Figure 4 shows the calculated surface pressurecoverage dependence assuming dipole-dipole interactions only between the colloidal particles at the oil-water interface. The simulations were carried out for point particles (dashed line) and for particles of radius 1.3 µm (full line). Since, as mentioned above, the dipole moment P is an unknown fitting parameter in the calculations, the absolute magnitude of the calculated surface pressure is arbitrary. Hence, the results shown in this figure are relative changes in the surface pressure Π, which were obtained by normalizing the calculated values of Π at the various coverages to the calculated value of Π for point particles at a coverage of 0.1 (i.e. the lowest coverage simulated in the MD calculations). As expected (and confirmed by the experimental observations6 shown in Figure 7), the surface pressure increases with increasing coverage. For low coverages, that is, large particle separations, the two sets of calculations assuming point particles and a particle radius of 1.3 µm are virtually
A N2
〈∑∑δ(r - r )〉 ij
(11)
i j*i
and shown in Figure 6, exhibits sharp, individual peaks at the nearest-neighbor HCP lattice constants for this structure, which confirms that the system is at least in a local minimum-energy configuration. Figure 7 shows the calculated surface pressurecoverage curves for monolayers consisting of R ) 1.3 µmsize particles at the oil-water interface assuming chargecharge (dashed line), charge-dipole (dotted line), and dipole-dipole (full line) interactions. Also shown in this figure, by the full circles, are the experimental results of Aveyard et al.6 As mentioned above, the surface charge density and the magnitude of the dipole moment are fitting parameters in the present calculations. We have therefore normalized the calculated surface pressure values in the left panel of Figure 7 to the experimental value of Π at a coverage of 0.55, whereas in the right panel the values were normalized to the experimental value of Π at a coverage of 0.3. (The values of the two fitting parameters, Cc and P, will be discussed in more detail in the following section.) There are several points that can be noted from Figure 7. First, it can be seen that each interaction mechanism gives rise to a characteristic surface pressurecoverage relationship. Second, comparing the experimental results with the calculated surface pressure curves shows that there are two regimes of interparticle interactions. From the left panel of Figure 7 it can be seen that the calculated surface pressure-coverage curve assuming dipole-dipole interactions only reproduces very well the experimental results at high coverages, while the right panel shows that charge-charge interactions only are sufficient to account for the observations at low coverages. We note that the simulations assuming point particles under dipole-dipole interactions (as, for example, shown in Figure 4) cannot reproduce the experimental data of Aveyard et al.6 This clearly demonstrates the importance of the inclusion of the size of the particles in the derivation
Surface Pressure of Colloidal Monolayers
Figure 8. Schematic diagram of a polystyrene particle at an oil-water interface. Surface charges, surface charge dipoles, and the nett dipole moment are also shown.
of the interaction energies for the simulation of micrometer-size particle systems. 5. Physical Interpretation We will now address possible physical interpretations of the observed particle interaction mechanisms. Robinson and Earnshaw5 and Aveyard et al.6 proposed that the particle surface, which is not molecularly smooth, will trap water at the particle-oil interface. This could be in the form of either small water droplets or a thin layer of water separating the particle surface from the oil phase. It is likely that the formation of such traces of water at the oil-particle interface arises from the rotation of the particles during the initial particle spreading process and/ or during the monolayer compression. The hydrophilic sulfate headgroups on the surface of the polystyrene particles are dissociated in contact with water, which, in turn, gives rise to surface charge dipoles. (This is schematically illustrated by the ellipsoids in Figure 8.) These surface charge dipoles are effectively screened in the aqueous phase. (For pure water the Debye length is ≈1 µm and decreases to ≈1 nm for a 0.1 M NaCl solution; solutions up to 1 M NaCl were employed in the experiments, while the particle separations were typically on the order of 3-14 µm for surface pressures between 50 mN/m and almost zero.6) However, in the oil phase the vector sum of the individual dipole moments gives rise to a net dipole moment (represented by the vector P in Figure 8). As mentioned in section 2, in the present calculations we have assumed that all colloidal particles in the monolayer have a net dipole moment which has, on average, the same magnitude and is oriented perpendicular to the oil-water interface. These assumptions are supported by the very regular lattice structure which is observed experimentally.6 In addition, Aveyard et al.6 attributed the observed long-range repulsions between the particles at the oil-water interface to Coulomb interactions arising from residual charges at the oilparticle interface. The origin and exact location of these surface charges is still unclear. However, it is possible that, perhaps also due to the rotation of the particles during the initial spreading process, some of the sulfate headgroups at the particle-oil interface lose their positive ion (either H+ or Na+). (This is schematically illustrated by the circles at the particle-oil interface in Figure 8.) The remaining positive ions either could be trapped near sulfate groups at the particle-water interface (as shown in Figure 8) or could be located in the aqueous phase near the particle-oil-water contact line. In either case, the positve charges are effectively screened in the aqueous phase and only the negative charges at the particle-oil interface give rise to the observed interparticle repulsions mediated by the oil phase. The present calculations are consistent with this interpretation, namely that it is
Langmuir, Vol. 17, No. 10, 2001 3107
dipole-dipole interactions which govern the short-range repulsion and charge-charge interactions which dominate the long-range repulsion between the polystyrene particles at the oil-water interface. We now return to a discussion of the two fitting parameters which were employed in the MD simulations of the colloidal monolayers, that is, the surface charge density Cc and the dipole moment P. From the results of the calculations, shown in Figure 7 and discussed in the previous section, we have argued that the high-coverage regime is dominated by dipole-dipole interactions, whereas the low-coverage regime is mainly governed by charge-charge interactions between the colloidal particles. Hence, we can determine Cc by adjusting its value in the calculations such as to give agreement with the experimental value of Π at low coverages. Similarly, P can be obtained by adjusting its value to give agreement between theory and experiment in the high-coverage regime. The values of the two fitting parameters thus obtained are Cc ) 2.69 × 10-4 C m-2 and P ) 1.65 × 10-20 C m. (For reasons of simplicity, we have for each case ignored the influence of the other interaction mechanisms. However, according to the relatively good fits in the two coverage regimes shown in Figure 7, the error introduced by this approximation is expected to be small.) We note that the maximal surface charge density and the contact angle Θow for particles at the oil-water interface are given in ref 6 as 7.7 µC cm-2 (stated by the manufacturer) and Θow ≈ 80° for the R ) 1.3 µm-size particles. Therefore, according to our calculations, about 0.4% of the total possible charge on the particle surface is required to account for the observed long-range repulsion through the oil phase. This value agrees remarkably well with the estimation by Aveyard et al.19 of around 1%. With regard to the dipole moment, if we (hypothetically) assume that the total charge on the surface area of the particle in the oil phase contributes to the dipole moment, that is,
q ) 2πR2(1 - cos Θow)7.7 µC cm-2 ) 6.76 × 10-13 C then the distance by which the two charges of the “net” dipole would need to be separated is (P/q )) 2.4 × 10-8 m. As indicated above, this value is only a rough estimate (and a lower limit), since it takes no cognizance of the facts that, for example, the surface charge dipoles have different directions and that only a fraction of the total surface charge density contributes to the formation of surface charge dipoles. However, we note that the obtained charge separation appears to be realistic in comparison with the particle size, that is, the calculated charge separation corresponds to 1.9% of the particle radius. 6. Conclusions (1) Molecular dynamics simulations have been presented of the compression of colloidal monolayers at the oil-water interface. The interaction energies between the colloidal particles employed in the simulations were derived for both point particles and particles of radius R. (2) A comparison of the calculated surface pressure values with the experimental results of Aveyard et al.6 showed that the observed short-range repulsion between the particles is governed by dipole-dipole interactions, while the long-range repulsion is mainly controlled by charge-charge interactions. The importance of the inclusion of the size of the particles in the derived interaction energies for micrometer-size particle monolayers was also demonstrated. (3) Values for the surface charge density and the surface dipole moment were obtained by fitting the calculations
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to the experimental results.6 The latter values were shown to be consistent with a physical interpretation involving traces of water at the oil-particle interface giving rise to surface charge dipoles and single surface charges interacting through the oil phase.
Sun and Stirner
Acknowledgment. The authors would like to thank Prof. Aveyard, Dr. Clint, and Prof. Hagston for valuable discussions. LA001574K
