6636
Langmuir 2005, 21, 6636-6641
Molecular Dynamics Simulation of the Structural Configuration of Binary Colloidal Monolayers T. Stirner*,† and Jizhong Sun†,‡ Department of Chemistry, University of Hull, Hull HU6 7RX, U.K., and Department of Physics, Dalian University of Technology, Dalian 116024, China Received February 14, 2005. In Final Form: April 21, 2005 Molecular dynamics simulations of binary colloidal monolayers, i.e., monolayers consisting of mixtures of two different particle sizes, are presented. In the simulations, the colloid particles are located at an oil-water interface and interact via an effective dipole-dipole potential. In particular, the influence of the particle ratio on the configurations of the binary monolayers is investigated for two different relative interaction strengths between the particles, and the pair correlation functions corresponding to the binary monolayers are calculated. The simulations show that the binary monolayers can only form two-dimensional crystals for certain particle ratios, for example, 2:1, 6:1, etc., while, for example, for a particle ratio of 7:1 the monolayers are found to be in a disordered, glassy state. The calculations also reveal that in analogy to the Wigner lattice the configurations are very sensitive to the relative interaction strength between the particles but not to the absolute magnitude of the interaction strength, even when particle size effects are taken into account. Consequently, it is argued that a comparison between the calculated configurations and actual binary particle monolayer systems could provide useful information on the relative interaction strength between large and small particles. Possible mechanisms giving rise to disparities in the interaction strength between large and small particles are described briefly.
1. Introduction Monolayers of colloidal particles at liquid-air and liquid-liquid interfaces have recently attracted a great deal of attention1-12 because of the large number of potential applications ranging from the food, agrochemical, and petrochemical industries to the pharmaceutical and cosmetics industries. Scientifically, monolayers of micrometer-size polystyrene latex and silica particles are of particular interest, since these particles can be observed directly using optical microscopy. Pieranski2 was the first to report a system of polystyrene particles trapped at the air-water interface, forming a single monolayer in a triangular phase. Highly ordered monodisperse particle monolayers are now being routinely fabricated,7-9,13 and a whole range of aspects, e.g., the compression-dependent surface pressure,8,9 colloidal particle aggregation,14 pointdefect diffusion mechanisms,15 etc., have been investi† ‡
University of Hull. Dalian University of Technology.
(1) Sheppard, E.; Tcheurekdjian, N. J. Colloid Interface Sci. 1968, 28, 481. (2) Pieranski, P. Phys. Rev. Lett. 1980, 45, 569. (3) Hurd, A. J.; Schaefer, D. W. Phys. Rev. Lett. 1985, 54, 1043. (4) Hurd, A. J. J. Phys. A: Math. Gen. 1985, 18, L1055. (5) Armstrong, A. J.; Mockler, R. C.; O’Sullivan, W. J. J. Phys.: Condens. Matter 1989, 1, 1707. (6) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. Lett. 1993, 71, 715. (7) Aveyard, R.; Clint, J. H.; Nees, D. Colloid Polym. Sci. 2000, 278, 155. (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) Quesada-Pe´rez, M.; Moncho-Jorda´, A.; Martı´nez-Lo´pez, F.; Hidalgo-A Ä lvarez, R. J. Chem. Phys. 2001, 115, 10897. (11) Ghezzi, E.; Earnshaw, J. C.; Finnis, M.; McCluney, M. J. Colloid Interface Sci. 2001, 238, 433. (12) Horozov, T. S.; Aveyard, R.; Clint, J. H.; Binks, B. P. Langmuir 2003, 19, 2822. (13) Stancik, E. J.; Gavranovic, G. T.; Widenbrant, M. J. O.; Laschitsch, A. T.; Vermant, J.; Fuller, G. G. Faraday Discuss. 2003, 123, 145. (14) Moncho-Jorda´, A.; Martı´nez-Lo´pez, F.; Gonza´lez, A. E.; HidalgoA Ä lvarez, R. Langmuir 2002, 18, 9183.
gated. Similarly, binary (bimodal) colloidal suspensions have been studied experimentally16-19 and theoretically.20-22 In particular, Ristenpart, Aksay, and Saville23 used an ac electric field to assemble planar superlattices in binary colloidal suspensions and observed triangular or square-packed arrays depending on the field frequency and relative particle concentrations. The main focus of these investigations was to explore the interaction mechanism between the colloidal particles, an understanding of which is of paramount importance for controlling the properties of liquid surfaces. The longrange order of the polystyrene-particle monolayers has been attributed to dipole-dipole repulsions arising from the asymmetric distribution of the counterion cloud around the particles.2,4 However, the recent experiments of Aveyard et al.,7-9 where particle monolayers at both the air-water and oil-water interfaces have been studied, cannot be easily explained by the existence of an electric double layer at the interface between the particles and the aqueous phase only. In short, for monolayers at the air-water interface it was found8 that with increasing electrolyte concentration in the aqueous phase the particles formed aggregates while the particles at the oilwater interface remained highly ordered. Prior to the experiments of Aveyard et al.,7-9 Robinson and Earnshaw24 had already proposed the existence of dipoles at the (15) Pertsinidis, A.; Ling, X. S. Nature 2001, 413, 147. (16) Dinsmore, A. D.; Yodh, A. G.; Pine, D. J. Phys. Rev. E 1995, 52, 4045. (17) Urban, C.; Schurtenberger, P. J. Colloid Interface Sci. 1998, 207, 150. (18) Crocker, J. C.; Matteo, J. A.; Dinsmore, A. D.; Yodh, A. G. Phys. Rev. Lett. 1999, 82, 4352. (19) Shenoy, S. S.; Sadowsky, R.; Mangum, J. L.; Hanus, L. H.; Wagner, N. J. J. Colloid Interface Sci. 2003, 268, 380. (20) Qin, K. D.; Zaman, A. A. J. Colloid Interface Sci. 2003, 266, 461. (21) Pierce, F.; Chakrabarti, A.; Fry, D.; Sorensen, C. M. Langmuir 2004, 20, 2498. (22) Varga, I.; Kun, F.; Pal, K. F. Phys. Rev. E 2004, 69, 030501. (23) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. Phys. Rev. Lett. 2003, 90, 128303. (24) Robinson, D. J.; Earnshaw, J. C. Langmuir 1993, 9, 1436.
10.1021/la050402q CCC: $30.25 © 2005 American Chemical Society Published on Web 05/27/2005
Binary Colloidal Monolayers
particle-air interface. A similar mechanism was also put forward to explain the amazing stability of the particle monolayers at the oil-water interface even for very high salt concentrations in the aqueous phase.25 These surface dipoles arise essentially from the dissociation of and subsequent counterion capture by the hydrophilic sulfate headgroups at the surface of the polystyrene particles when they come into contact with water;26 i.e., a thin water film may remain from the particle spreading process at the particle-air or particle-oil interface. This water film could be either continuous over the whole interface or in the form of tiny droplets trapped at the particle surface because of microroughness. The difference between the behavior of the particle monolayers at the air-water and oil-water interfaces has been mainly attributed to the different contact angles at the two interfaces.8 However, it is also clear that such a water film as described above would easily evaporate (at least partially) at the particleair interface, while at the particle-oil interface it would be relatively stable. A combination of these two effects could thus also provide a plausible explanation for the observed difference in the particle aggregation behavior at the air-water and oil-water interfaces. In our earlier work, we have employed the molecular dynamics technique to study the surface pressure,25 lattice, and drag forces on optically trapped colloidal particles26 and solid-to-solid phase transitions27 occurring in unary particle monolayers. Our simulations showed that an effective dipole-dipole interaction between the particles can reproduce successfully the observed8,28 two-dimensional, triangular configurations, the hexagonal-to-rhombohedral phase transition under anisotropic compression, and the lattice force on an optically trapped colloidal particle in a moving particle monolayer. Interestingly, we also found that the surface pressure is relatively insensitive to the exact configuration of the monolayer as long as the particle coverage is unchanged. Consequently, the observed configurations (including the derived pair correlation functions, global orientational order parameters, etc.) of the monolayers consisting of monodisperse particles are of limited use in gaining information on the particle interaction mechanism. In the present work we are therefore concerned with the structural properties of binary colloidal monolayers, i.e., monolayers consisting of a mixture of large and small particles. The main motivation for this work stems from the idea that a comparison between experimental and simulated configurations (in particular the large-large, large-small, and small-small particle separations) may provide us with useful information on the relative particle interaction strengths. In what follows, we will investigate several different particle ratios within two different interaction regimes (strong and weak). Here, we will focus on polystyrene particles trapped at an interface formed by an oil phase and an aqueous phase, where we know from our earlier work26 that an effective dipole-dipole potential is an excellent approximation of the interaction between two polystyrene particles. Consequently, we will employ this same effective dipole-dipole potential in the present computer simulations. (25) Sun, J. Z.; Stirner, T. Langmuir 2001, 17, 3103. (26) Sun, J. Z.; Stirner, T. J. Chem. Phys. 2004, 121, 4292. (27) Sun, J. Z.; Stirner, T. Phys. Rev. E 2003, 67, 051107. (28) Aveyard, R.; Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Horozov, T. S.; Neumann, B.; Paunov, V. N.; Annesley, J.; Botchway, S. W.; Nees, D.; Parker, A. W.; Ward, A. D.; Burgess, A. N. Phys. Rev. Lett. 2002, 88, 246102.
Langmuir, Vol. 21, No. 14, 2005 6637
Figure 1. Schematic diagram of the polystyrene particle arrangement at the oil-water interface (ø on the order of micrometers) and the interaction mechanisms acting through the oil phase (surface charge dipoles formed from hydrophilic sulfate headgroups and a diffuse cloud of fully hydrated counterions; a small fraction of the sulfate groups lose their counterion and form monopoles) and through the aqueous phase (DLVO interaction is negligible because of a very small Debye length, 1/κ on the order of angstroms). The net effective dipole moment P is the vector sum of all surface charge dipole moments at the particle-oil interface.
2. Computer Simulation Details In ref 25 we have derived an analytical expression for the interaction energy between two parallel effective dipoles of moment P. The formation of the effective dipole moment at the particle-oil interface is shown schematically in Figure 1. Each dipole was assumed to be perpendicular to the oil-water interface and located on a particle of radius R, with the center-to-center separation between the particles being denoted by r. The final equation is given by
E)
P2 [2 ln r - ln(r + 2R) - ln(r - 2R)] (1) 16πrR2
where is the permittivity of the dielectric medium. The behavior of eq 1 at long range approaches that of a pointdipole potential (i.e., P2/(4πr3)) and deviates from the latter by less than 15% for r J 4R. However, in contrast to a simple point-dipole potential, eq 1 behaves like a hard-sphere potential for r f 2R. Following the same method as described previously,25 it is straightforward to derive the corresponding expression for the interaction energy between two particles of radii R1 and R2, i.e.,
E)
[
]
r2 - (R1 - R2)2 πp1p2R1R2 ln 2 4r r - (R1 + R2)2
(2)
where p1 (p2) is the magnitude of the dipole moment surface density of particle 1 (particle 2). We define the magnitude of the effective dipole moments, P1 and P2, of particles 1 and 2 as
P1 ) 2πR12(1 - cos Θ1)p1 and
P2 ) 2πR22(1 - cos Θ2)p2
(3)
where Θ1 (Θ2) is the contact angle of particle 1 (particle 2). If the contact angle is the same for the two particles, the effective dipole moment is simply proportional to the product of p and R2. p depends on the surface charge density, the degree of dissociation of the sulfate head-
6638
Langmuir, Vol. 21, No. 14, 2005
groups, the type of trapped counterion, and also the properties of the dielectric environment. For two different particle sizes there are three possible combinations of interactions between the particles: (i) interactions between two large particles, (ii) interactions between a large and a small particle, and (iii) interactions between two small particles. These combinations lead to three different interaction coefficients that we define as Cl ) [Pl/(2Rl)]2, Cls ) PlPs/(4RlRs), and Cs ) [Ps/(2Rs)]2, where Pl (Rl) represents the effective dipole moment (radius) of a large particle and Ps (Rs) denotes the effective dipole moment (radius) of a small particle. These three coefficients will be treated as free parameters in the simulations. The molecular dynamics technique was primarily developed to investigate time-dependent and time-average properties of atomic and molecular systems.29 In addition, however, it can also be used to search for the global energy minimum of a complex system. In the present work we will utilize the molecular dynamics technique to investigate the energy surface of the binary colloidal monolayers. To make the water-oil-particle monolayer system tractable on a computer, only the interactions between the colloidal particles are evaluated in the molecular dynamics simulations, while the oil phase is described by its static dielectric constant in the particle interaction energies (only the interactions between the particles through the oil phase are considered; from the experiments8,28 on monodisperse particle monolayers, which were carried out at varying salt concentrations in the aqueous phase, we know that the interactions between the particles through the aqueous phase can be neglected because of large electrostatic screening effects). The oilwater interface is assumed to be planar, since we know from the experiments8 that the colloidal particles are trapped in a deep surface potential well and that monolayer folding and/or corrugations only occur above the monolayer collapse pressure. In the present study the particles are large enough not to be affected significantly by Brownian motion. The molecular dynamics simulations are performed for 121 large particles, and the number of small particles is calculated from the product of the chosen particle ratio and the number of large particles. The colloidal particles are contained in a rectangular simulation cell with periodic boundary conditions. Both the system size and the dimensions of the simulation cell (side ratio 2/x3) were chosen such that shear stresses are negligible. Unless stated otherwise, the initial distribution of the large particles is hexagonal with the small particles being distributed evenly between the large particles. The particle velocities are sampled randomly from a MaxwellBoltzmann distribution (T ) 298 K).30 We use the petitcanonical ensemble with a Nose´-Hoover thermostat and the velocity Verlet algorithm30 to solve Newton’s equations of motion numerically. The ratio of the potential to kinetic energy for the monolayers investigated was chosen such that U/(NkBT) ) 5 × 102 at equilibrium. The monolayers are equilibrated with a time step of 1 ns during the first 40 000 steps and with a time step of 4 ns during the subsequent 100 000 steps. The particle parameters used in the simulations are typical for real polystyrene particles,8,28 i.e., a large particle diameter of 2.7 µm, a small particle diameter of 0.89 µm, a polystyrene mass density of 1.06 g cm-3, and a surface charge density of 8.9 µC cm-2. The convergence of the average kinetic and potential (29) Leach, A. R. Molecular Modelling: Principles and Applications; Longman: Singapore, 1996. (30) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1992.
Stirner and Sun
energies was used as a criterion for the equilibration of the system. We have carried out a wide variety of simulations for different system sizes, initial distributions, and energy ratios and find that all qualitative features are robust. To characterize the structure of the colloidal monolayers quantitatively, we have evaluated the pair correlation function30
g(r) )
A N
N N
∑ ∑δ(r - rij)〉 2 i)1 j*i 〈
(4)
for the entire binary monolayer as well as for the large and small particle sublattices. Here, A is the area of the simulation cell, N is the number of particles, and rij()ri - rj) is the separation between particles i and j. Equation 4 gives the probability of finding a pair of particles a distance r apart, relative to the probability for a completely random distribution at the same particle density. 3. Results and Discussion In what follows, we will first investigate how the monolayer configuration depends on the particle ratio (defined as the number of small to large particles). The interaction energy described by eq 2 is employed in the simulations, where the interaction coefficients Cl and Cs are chosen such that Cl/Cs ) (Rl/Rs)3. It seems reasonable to assume that the degree of dissociation of the sulfate headgroups and the type of trapped counterion are the same for the large and small particles. Therefore, the ratio of the interaction coefficients chosen here can be interpreted in the sense that the dielectric environment is different for the two particle types, thus giving rise to a larger dipole moment surface density for the large particles compared to that of the small particles (we will come back to this point in the next section). From our earlier simulations25,27 and the experimental studies2,5,8 it is well-known that monolayers of monodisperse polystyrene particles form a hexagonal structure at the air-water and oil-water interfaces. The potential energy surface of a perfect hexagonal structure consisting of N identical particles has 2N stable points (located at each center of a triangle) and 3N metastable points (located at each saddle point between two vertexes). The stable points have a 3-fold symmetry axis, and the metastable points have a 2-fold symmetry axis. However, for a binary particle monolayer, the energy surface is more complicated. In fact, the incorporation of small particles into the monolayer reduces the energy barrier between two stable points. Following these symmetry arguments, we can expect certain particle ratios, i.e., 2:1, 6:1, 8:1, etc., for which a binary particle monolayer is more ordered than for other ratios. For instance, at a particle ratio of 1:1, the small particles are easily trapped at the centers of the triangles formed by the large particles. The final configuration depends on the initial distribution of the small particles because there are twice as many stable energy minima as there are small particles. Consequently, in what follows, we will first investigate a particle ratio of 2:1 in more detail. 3.1. Particle Ratio 2:1. Figure 2 shows an equilibrated configuration for a particle ratio of 2:1. The average separation between two large particles is ∼6 µm. As can be seen from Figure 2, the small particles occupy the energy minima on a hexagonal sublattice between an almost perfect triangular structure formed by the large particles. Consequently, it is the large particles that control the overall long-range nature of the configuration, while
Binary Colloidal Monolayers
Figure 2. Equilibrated configuration for a particle ratio of 2:1 (strong interaction regime). The large (small) particles are represented by large (small) circles. The average separation between the large particles is ∼6 µm.
Figure 3. Pair correlation function corresponding to the configuration of Figure 2.
Figure 4. Equilibrated configuration for a particle ratio of 6:1 (strong interaction regime). The average separation between the large particles is ∼6 µm.
the small particles (occupying all stable interstitial sites) do not influence significantly the crystal structure of the monolayer. Figure 3 shows the pair correlation function corresponding to the configuration of Figure 2. As can be seen, the pair correlation function has very sharp peaks reflecting the good long-range order of the monolayer. Its first peak is generated by pairs of large and small particles and by small nearest-neighbor particles separated by ∼6/ x3 µm, the second peak at r ≈ 6 µm is due to pairs of large nearest-neighbor particles and small second-nearestneighbor particles, whereas the third peak at ∼12/x3 µm is created by pairs of small particles with a large particle in between and by large and small particles with a small particle in between. 3.2. Particle Ratio 6:1. We now consider a particle ratio of 6:1 in more detail. An equilibrated configuration for this particle ratio is shown in Figure 4, where the center-to-center separation between two large particles is ∼6 µm. From Figure 4 it can be seen that the small particles are distributed along an energy-favored corridor around each large particle. If we examine a triangle formed by three large particles closely, we can see that a smaller (inverted) triangle consisting of three small particles is located at the center of the large triangle. The configuration is not entirely “frozen” because the system has some kinetic energy; however, the movements of the small
Langmuir, Vol. 21, No. 14, 2005 6639
Figure 5. Pair correlation function corresponding to the monolayer of Figure 4. The full line is for the entire monolayer, and the dotted line for the small particles only.
particles (due to thermal motion around the equilibrium state) are severely restricted by the potential energy barriers generated by the large particles and no diffusion is taking place. The pair correlation function corresponding to the monolayer of Figure 4 is shown by the full line in Figure 5. Also shown in Figure 5, by the dotted line, is the pair correlation function for the small particles only. It is apparent from this figure that the first peak of the pair correlation function is created by small nearest-neighbor particles only, while most other peaks are generated by combinations of large and small particles. While there is a small amount of disorder in the lattice, the configuration shown in Figure 4 corresponds to that of a good crystal with a unit cell composed of one large particle and six small particles (see, for example, a rhombus formed by four large particles). We will now investigate how the relative interaction strength between the particles affects the configuration of the binary monolayers. As mentioned above, for the monolayer configurations presented so far, we have chosen the interaction coefficients for the large and small particles such that they are described by the ratio Cl/Cs ) (Rl/Rs)3 (we will refer to this case as the “strong interaction regime”, i.e., the interactions between the large particles are stronger compared to the interactions between the small particles even when particle size effects are taken into account). In what follows we will present the resulting configurations for an interaction coefficient ratio of Cl/Cs ) (Rl/Rs)2 (we will refer to this case as the “weak interaction regime”; i.e., the interactions between the large particles scale exactly in proportion to their size when compared to the interactions between the small particles). To compare directly the simulation results for these two regimes, we have scaled the potential energy for the weak interaction regime to be approximately equal to that of the strong interaction regime. Figure 6 shows an equilibrated monolayer configuration for the weak interaction regime at a particle ratio of 6:1. In comparison with Figure 4 (obtained under the strong interaction regime), it is apparent that the perfect crystal structure is lost for the weak interaction regime. This is mainly due to the fact that the large particles are unable to form a perfect triangular lattice. Nevertheless, the largeparticle sublattice still exhibits a relatively good longrange triangular order. The pair correlation function corresponding to the configuration of Figure 6 is shown by the full line in Figure 7. Also shown in Figure 7, by the dotted line, is the pair correlation function of Figure 5, i.e., for the corresponding strong interaction regime. As can be seen, the weaker interactions between the large particles cause the first peak of the pair correlation function (associated with small
6640
Langmuir, Vol. 21, No. 14, 2005
Figure 6. Equilibrated configuration for a particle ratio of 6:1 (weak interaction regime). The average separation between the large particles is ∼6 µm.
Figure 7. Pair correlation functions corresponding to the monolayers of Figure 6 (full line, weak interaction regime) and Figure 4 (dotted line, strong interaction regime; see Figure 5).
nearest-neighbor particles only) to shift from ∼1.5 to ∼1.75 µm, while in the vicinity of the second peak for the strong interaction regime (at ∼3 µm) two peaks emerge (mainly because of large and small first-nearest-neighbor particles and large and small second-nearest-neighbor particles). Furthermore, there is no distinct peak at 6 µm for the weak interaction regime (rather a broadend band covering the locations of the higher order peaks) reflecting the decrease in long-range order of the large-particle sublattice compared to the strong interaction regime. We have repeated the simulations for lower particle coverages (i.e., larger particle separations of ∼12 µm) and, although the magnitude of the interaction between the
Stirner and Sun
particles is weaker, have found no obvious differences between the configurations at different particle coverages. This demonstrates that the configurations are not sensitive to the absolute strength of interaction in the range of coverages investigated (only to the relative interaction strengths) and is analogous to the findings obtained for the Wigner lattice.31,32 The differences in the configurations between strong and weak interaction regimes are even more pronounced for a particle ratio of 7:1. This is discussed in the following section. 3.3. Particle Ratio 7:1. We have repeated the simulations for a particle ratio of 7:1. Symmetry arguments show that for such a particle ratio a crystalline configuration cannot be obtained irrespective of the initial distribution of the small particles (i.e., it is impossible for the large particles to form a perfect triangular lattice because the small particles cannot be evenly distributed among the large particles). This is illustrated in Figure 8, which shows two equilibrated configurations for a particle ratio of 7:1 obtained from a random initial particle distribution. Figure 8a corresponds to the strong, and Figure 8b corresponds to the weak interaction regime. The simulation cell area is 156 × 110 µm2, and the average separation between the large particles is ∼12 µm. As can be seen from this figure, there is a marked difference in the resulting configurations for the two interaction regimes. For the strong interaction regime (Figure 8a), there is still some degree of long-range triangular order in the large-particle sublattice. Many large particles form (sometimes considerably distorted) hexagons, and some small particles form triangles located at the center of a large-particle triangle. However, for the weak interaction regime (Figure 8b) the long-range triangular order of the large-particle sublattice is lost. Instead, some of the small particles are able to form rafts with relatively good short-range triangular order. We note that to test the stability of these monolayers, we have employed a simulated annealing technique (i.e., heating of the monolayer to ∼600 K followed by equilibration and quenching back to 298 K) and have found neither particle demixing nor in fact any significant changes in the corresponding pair correlation functions before and after annealing. Experimental configurations are, in general, difficult to reproduce exactly on a computer because of the complications in achieving the global energy-minimum configuration (both experimentally and numerically). However, the pair correlation functions can also be obtained directly from the experimental monolayers.5 For binary monolayers, the first peak of the pair correlation
Figure 8. Equilibrated configurations resulting from a random initial particle distribution for a particle ratio of 7:1 and (a) the strong and (b) the weak interaction regimes.
Binary Colloidal Monolayers
Langmuir, Vol. 21, No. 14, 2005 6641
of the first peak is also reflected in the second peak (see top panel), while for large and small particles, the second peak in the pair correlation function is rather insensitive to the relative interaction strength (see bottom panel). These findings apply almost irrespective of the quality of the monolayer and can therefore be utilized to gain information on the relative interaction strength between the particles in binary monolayers, e.g., via the comparison of theoretical and experimental pair correlation functions. 4. Conclusions Figure 9. Pair correlation functions corresponding to the monolayers of Figure 8. The pair correlation functions for small particles (top panel) and between large and small particles (bottom panel) are shown separately. The full (dotted) line corresponds to the weak (strong) interaction regime.
function is determined only by the separation between small nearest-neighbor particles, and at constant coverage and interaction strength, the peak associated with large and small nearest-neighbor particles is almost independent of the configuration as long as the monolayer is close to equilibrium. Figure 9 shows the pair correlation functions corresponding to the configurations of Figure 8 for small particles only (top panel) and for large and small particles only (bottom panel). The full (dotted) lines in Figure 9 correspond to the weak (strong) interaction regime. As can be seen from this figure, the first peaks are quite well-defined (despite the relatively poor longrange order, in particular of the large particles in Figure 8b) and occur at different values of r for the two interaction regimes. Where the peaks occur depends on the relative interaction strength. For the strong interaction regime, the first peak formed only by small particles appears at a smaller value of r (see top panel of Figure 9), while the first peak formed by large and small particles occurs at a larger value of r (see bottom panel of Figure 9). In addition, for small particles only, this shift in the position (31) Goldoni, G.; Peeters, F. M. Phys. Rev. B 1996, 53, 4591. (32) Schweigert, I. V.; Schweigert, V. A.; Peeters, F. M. Phys. Rev. Lett. 1999, 82, 5293.
Molecular dynamics simulations of binary colloidal monolayers were presented. In the simulations the particles are assumed to be located at an oil-water interface and interact via an effective dipole-dipole potential. The influence of the ratio of small to large particles on the configurations of the mixed monolayers was investigated for two different relative interaction strengths. The molecular dynamics simulations showed that for particle ratios of 2:1 and 6:1 the binary monolayers can form a two-dimensional crystal, similar to the novel colloidal crystalline states observed by Reichhardt and Olson33 and Brunner and Bechinger34 on two-dimensional periodic substrates. For a particle ratio of 7:1 the monolayers were found to be in a disordered, glassy state. The calculations also revealed that, in analogy to the Wigner lattice, the configurations are very sensitive to the relative interaction strength between the particles but not to the absolute magnitude of the interaction strength. Possible mechanisms that could give rise to a marked difference in the interaction strengths between the large and small particles, respectively, are (i) a disparity in the dielectric environment between the large and small particles, e.g., differences in the properties of the water film at the particle-oil interface (i.e., average film thickness, coverage, etc.), (ii) a variation in the contact angle with particle size, and (iii) a combination of (i) and (ii). LA050402Q (33) Reichhardt, C.; Olson, C. J. Phys. Rev. Lett. 2002, 88, 248301. (34) Brunner, M.; Bechinger, C. Phys. Rev. Lett. 2002, 88, 248302.
