Langmuir 2007, 23, 5445-5451
5445
Modeling the Structure Formation of Particulate Langmuir Films: the Effect of Polydispersity Attila Agod,† Norbert Nagy,‡ and Zolta´n Ho´rvo¨lgyi*,† Budapest UniVersity of Technology and Economics, Department of Physical Chemistry, H-1521, Budapest, Hungary, and Research Institute for Technical Physics and Materials Science, P. O. Box 49, H-1525 Budapest, Hungary ReceiVed NoVember 30, 2006. In Final Form: February 14, 2007 Two-dimensional molecular dynamics computer simulation has been developed to model the compression of Langmuir films composed of spherical nanoparticles with arbitrary size distribution. We demonstrate that the usual assumption in the determination of interparticle potentials from the surface pressure vs area isotherms (i.e., monodisperse particles in perfect hexagonal order) leads to a systematic overestimation of the characteristic length of the interaction. On the basis of the results of the simulation, we propose a correction method to improve the traditional way of determining the interparticle potentials. We use the corrected particle-particle interactions to explore the correlation between the broadness of the size distribution and several structural parameters (decay length of pair-correlation function, global orientational order parameter, mean, and standard deviation of number of neighbors). Due to the uniaxial compression and the stiffness of the particulate layer, the surface pressure is not a scalar field. We investigate the effect of polydispersity on the anisotropy and the fluctuation of the surface pressure tensor in Langmuir films during uniaxial compression.
Introduction In the colloid chemical approach to nanotechnology, molecules and nanoparticles are the building blocks of the new materials. The macroscopic properties of these nanostructures are determined not only by the characteristics of the individual particles but also by their spatial order. Thus, if the mechanisms of structure formation are known, their physical and chemical properties can be designed in a more systematic way. By means of a computer simulation, the multidimensional space of input parameters can be easily explored; thus, correlations between the measurable quantities and the microscopic mechanisms are much easier to reveal. Structuring a material on the nanoscale can provide new physical or chemical characteristics, e.g., modulating the dielectric permittivity on the length scale comparable to the wavelength of light results in photonic crystals1 or antireflective coatings.2 The electronic band structure of nanometer-sized particles may differ from the macroscopic bulk characteristics due to sizequantized effects.3 The relatively simple and inexpensive Langmuir-Blodgett (LB) technique allows us to exploit these novel properties on the macroscopic scale. By fabricating multilayers from the films of various types of nanoparticles, optical,4,5 electro-optical,6 catalytic,7 magnetic,8,9 and wetting10 * To whom correspondence should be addressed. E-mail: zhorvolgyi@ mail.bme.hu. † Budapest Institute of Technology and Economics. ‡ Research Institute for Technical Physics and Materials Science. (1) Yablonovitch, E. Phys. ReV. Lett. 1987, 58, 2059-2062. (2) Dea´k, A.; Sze´kely, I.; Ka´lma´n, E.; Keresztes, Zs.; Kova´cs, A. L.; Ho´rvo¨lgyi, Z. Thin Solid Films 2005, 484, 310-317. (3) Brus, L. E. J. Chem. Phys. 1984, 80, 4403-4409. (4) Dea´k, A.; Bancsi, B.; To´th, A. L.; Kova´cs, A. L.; Ho´rvo¨lgyi, Z. Colloids Surfaces A: Physicochemical Eng. Asp. 2006, 278 (1-3), 10-16. (5) Szekeres, M.; Kamalin, O.; Schoonheydt, R. A.; Wostyn, K.; Clays, K.; Persoons, A.; De´ka´ny, I. J. Mater. Chem. 2002, 12, 3268-3274. (6) Achermann, M.; Petruska, M. A.; Crooker, S. A.; Klimov, V. I. J. Phys. Chem. B 2003, 107, 13782-13787. (7) Takahashi, M.; Natori, H.; Tajima, K.; Kobayashi, K. Thin Solid Films 2005, 489, 205-214. (8) Kang, Y. S.; Lee, D. K.; Stroeve, P. Thin Solid Films 1998, 327-329, 541-544.
properties of materials can be combined. To gain a better understanding of the structure formation of nanoparticulate Langmuir films, we have developed a computer simulation that permits the modeling of film balance experiments of partially wettable, spherical nanoparticles. Several papers were published concerning computer simulations of particulate Langmuir films in the past decade. Sun and Stirner developed a two-dimensional molecular dynamics simulation to model monolayers consisting of polystyrene particles trapped at a water-oil interface and interacting via dipole-dipole potentials.11 According to their results, the particles formed a domain structure and the originally hexagonal domains transformed into rhombohedral ones due to the anisotropic compression. They also modeled the structure formation of bidisperse systems,12 and found that binary layers can only form two-dimensional crystals for certain particle ratios (e.g., 2:1, 6:1) and other ratios resulted in disordered, glassy states. Pugnaloni et al. used the Brownian dynamics technique to model the compression of monodisperse particles in a Langmuir trough.13 The particles were allowed to move perpendicular to the fluid-liquid interface, and this way they could follow the evolution of the film even after the onset of the collapse. The effects of polydispersity on the order-disorder transition in colloidal systems were investigated by several authors,14 but with respect to particulate Langmuir films, we have not found relevant papers. The interactions between particles trapped at the liquid-fluid interface have been intensively studied, e.g., ref 15. The interactions can also be estimated from the surface pressure vs (9) Lee, D. K.; Kang, Y. S. Colloids Surf., A 2005, 257-258, 237-241. (10) Tsai, P. S.; Yang, Y. M.; Lee, Y. L. Langmuir 2006, 22, 5660-5665. (11) Sun, J.; Stirner, T. Langmuir 2001, 17 (10), 3103-3108. (12) Stirner, T.; Sun, J. Langmuir 2005, 21, 6636-6641. (13) Pugnaloni, L. A.; Ettelaie, R.; Dickinson, E. Langmuir 2004, 20, 60966099. (14) Phan, S. E.; Russel, W. B.; Zhu, J.; Chaikin, P. M. J. Chem. Phys. 1998, 108 (23), 15. (15) 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 (24), 246102
10.1021/la063481u CCC: $37.00 © 2007 American Chemical Society Published on Web 04/04/2007
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Figure 1. Surface pressure (Π) vs area (A) isotherm of 44 nm diameter Sto¨ber silica particles at water-air interface. Ac denotes the contact cross-sectional area.
area isotherms by assuming monodisperse particles in perfect hexagonal order during compression and that only first-neighbor particle-particle (p-p) interactions are significant.16,17 The first goal of this work is to show that these assumptions lead to the overestimation of the range of p-p interactions. We will assess the systematic error of the traditional method and show the corrected interparticle potentials for the system of 44 nm Sto¨ber silica particles at the water-air interface. In practice, nanoparticles are never perfectly monodisperse, and the difference in diameters can have a significant effect on the final structure of the film. In this work, we will measure the polydispersity as the ratio of the standard deviation and the mean of the particle diameters. Our second goal is to explore the effect of the polydispersity of particles on the structure of their Langmuir film. We will investigate the dependence of several structural parameters (decay length of pair-correlation function, orientational order parameter, mean, and standard deviation of number of neighbors) on the polydispersity. The packing of the particles is not perfectly isotropic in a film balance because the barrier (or barriers) can push them only in one direction, i.e., the compression is uniaxial. Aveyard et al. showed experimentally that this anisotropy induces hexagonal to rhombohedral transition in the domains of Langmuir films.18 The restructuring of the particles is not continuous during compression due to the stiffness of the structure; hence, the surface pressure can depend on the direction in the film. We will make an attempt to find interrelation between the broadness of the size distribution, the fluctuation of the local surface pressure tensor, and the mechanisms of the restructuring in the compressed monolayer. In the next section of this paper, we give a short description of the simulation. Later, we present the correction method for the determination of interparticle potentials. After applying the corrected interactions for the system of Sto¨ber silica particles at water-air interface, we analyze the structure of the simulated particulate film. Then, from a more general aspect, we present the relationship between the polydispersity and several structural parameters. Finally, we reveal the correlation between the variation of surface pressure tensor and the mechanisms of structure formation. Simulation Details In the 2D model, disks correspond to spherical particles. We applied molecular dynamics technique, where pairs of particles can (16) Clint, J. H.; Quirke, N. Colloids Surf., A 1993, 78, 277. (17) Sheppard, E.; Tcheurekdjian, N. Kolloid Z. Z. Polym. 1968, 225, 162. (18) Aveyard, R.; Clint, J. H.; Nees, D.; Quirke, N. Langmuir 2000, 16, 88208828.
Agod et al. interact via arbitrary central potential. The interaction depends only on the distance between the rims of the disks; hence, it is independent of their size. Arbitrary size distribution can be set, and the disks can move in a rectangular area with periodic boundary conditions. In a compression step, two parallel sides of the rectangle are decreased, imitating the motion of the barrier in a real film balance, and the positions of the particles are rescaled in the direction of movement. After a compression step, the system is allowed to relax until every macroscopic parameter (temperature, pressure, average interaction energy, etc.) begins its natural fluctuation around its equilibrium value. To integrate the equation of motion of the system consisting of 2000 particles, we applied the leapfrog method with time steps of 10-11 s. Between two compression steps, the system relaxed for about 5000 time steps, and the whole simulated experiment consisted of 4000 compression steps. Since real experiments are carried out at constant temperature, the velocity of particles (Vi) has to be rescaled in every compression N (1/2)miV2i constant (where mi is the mass of the ith step to keep ∑i)1 particle). The surface pressure of the layer can be derived from the virial theorem.19 Π)
1 AD
N
〈
1
∑m V 〉 + AD 〈∑BF br 〉 2 i i
i)1
ij ij
(1)
(i,j)
where b rij ) b ri - b rj is the separation between the centers of mass of particles i and j, (i,j) are the interacting pairs, B Fij is the force between particles i and j, A is the total area of the layer and D is the dimension of the system.
Results and Discussion Determination of Particle-Particle Interactions. If the particles at the liquid-fluid interface form a gas film or their layer is just weakly cohesive, the repulsive part of the interaction energy can be estimated from the surface pressure vs area (ΠA) isotherms by assuming that monodisperse particles are packing in perfect hexagonal order. These assumptions are quite strict: in most cases, real particles have significant polydispersity, and due to the initially random structure and the continuous uniaxial compression, the layer is not in perfect hexagonal order even in the case of ideally monodisperse particles. In this section, we are going to show how the systematic error of these assumptions can be corrected in the case of 44 nm diameter Sto¨ber silica particles at the water-air interface. We analyzed the Langmuir films of 44 nm Sto¨ber silica in an earlier work, where we demonstrated the transmission electron microscopy images and the size distribution of the particles.20 The distribution of particle diameters can be approximated by a normal distribution with 44 nm expected value (d) and 8 nm standard deviation (σ). From the second moment of the distribution the average cross section (A h ), and from the third moment the average particle volume (V h ) can be determined. The greater the polydispersity, the more significant the difference between the volume of an average diameter sphere and the average volume, and the same holds in respect of cross-sectional areas. To determine the interaction potential, we started out from the Π-A isotherm shown in Figure 1. (Detailed description of the experiment can be found in ref 20.) The number of particles in the layer (N) can be derived from the total mass of the spread particles (N ) mtot/(FV h ), where F ) 2060 kg/m3 is the density of Sto¨ber silica21). Assuming that every particle has A h cross-sectional area, the layer is hexagonally (19) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1986. (20) Agod, A.; Tolnai, Gy.; Esmail, N.; Ho´rvo¨lgyi, Z. Progr. Colloid Polym. Sci. 2004, 125, 54-60. (21) Kabai-Faix, M. Magy. Ke´ m. Foly. 1996, 102, 33-41.
Modeling the Structure Formation of Langmuir Films
Langmuir, Vol. 23, No. 10, 2007 5447
Figure 2. Interaction potential determined from the Π-A isotherm demonstrated in Figure 1.
ordered and only the first neighbor interactions are significant, the interaction energy can be expressed as a function of surface area per particle (A1 ) A/N), (eq 2):
V(A1) ) -
1 3
∫AA /N Π(A1′) dA1′ 1
∞
(2)
The distance between the rims of two neighboring particles can be determined from A1, (eq 3); thus, the interaction energy can be expressed as a function of distance between the particle surfaces (rss).
rss ) 2‚
(x
A1
2x3
-
x) A h π
(3)
The interaction potential energy derived from the Π-A isotherm is shown in Figure 2 on a semilogarithmic plot. We found that exponential decay can be fitted before the onset of the collapse. Levin et al. reported similar interaction potentials for particles trapped at the oil-water interface.22 Thus, as a first approximation the interparticle potential is
V(rss) ) V0 e-rss/Λ
(4)
where V0 ) 3.81 × 10-15J and Λ ) 0.55 nm. Using the aforementioned model assumptions, the range of p-p interactions is systematically overestimated because a real particulate layer provides smaller coverage at a given surface pressure than an ideal monodisperse hexagonal array. The smaller coverage corresponds to larger average interparticle distances; thus, the calculated p-p potential decays at larger particle separations. The two parameters of the fitted exponential function include the systematic errors caused by the assumptions. For a given size distribution, these errors can be assessed and accordingly corrected by means of the simulation. The leading idea of the correction is that the error of a method can be revealed if it is applied in series. The steps of the algorithm are 1. determine the parameters of the potential from the experimental isotherm assuming monodisperse particles in hexagonal order (V0,1, Λ1) 2. employ this interaction in the simulation with the real size distribution, with initially randomly spread particles and applying uniaxial compression 3. determine the parameters of the potential from the simulated Π-A isotherm assuming monodisperse particles in hexagonal order (V0,2, Λ2 (22) Levine, S.; Bowen, B. D.; Partridge, S. J. Colloids Surf. 1989, 38, 325343.
Figure 3. Series of the parameters of the p-p interaction potentials (a, V0; b, Λ) and their extrapolation to zero evaluation step.
4. repeat (2) and (3), i.e., employ the last determined interaction in the simulation with realistic conditions, and determine the potential using the model assumptions (V0,3, Λ3, ...). As can be seen, the systematic error caused by the original model assumptions is accumulating in the series of parameters of the calculated interaction potentials. In Figure 3 we demonstrate these parameters as a function of evaluation steps for 44 nm Sto¨ber silica system. Both series are monotonous; thus, we can extrapolate the values to zero evaluation step and this way eliminate the systematic error of the traditional method where monodisperse spheres are assumed in perfect hexagonal order. In the following sections of the paper, we use the interaction potential energy determined by this new method (eq 5).
V(rss) ) 2.65 × 10-15J e-rss/0.46nm
(5)
Using the corrected potential (eq 5) in the simulation, the contact cross sectional area (Ac) determined from the simulated Π-A isotherm agrees within 0.5% with Ac determined from the original experimental isotherm, whereas using the uncorrected potential (eq 4) in the simulation results in 4% larger contact cross-sectional area. Structure of the Langmuir Film of 44 nm Sto1 ber Silica Particles. In this subsection, we analyze the simulated structure of the Langmuir film consisting of 44 nm diameter Sto¨ber silica particles. We applied the potential given by eq 5 between the particles and set the size distribution to match the original sample. In Figure 4 the structure of the modeled particulate layer at 10 mN/m surface pressure is demonstrated. On the right-hand side of the image in Figure 4, the particles are colored according to the number of their neighbors. Two particles are considered to be neighboring if there is direct interaction between them, in practice if the interaction energy defined by eq 5 is greater than kT. The demonstrated structure is obviously not hexagonal. The average number of neighbors is 5.07, which is significantly less than 6, the number characterizing the hexagonal order. The simulated monolayer is sparser than a perfect hexagonal array of monodisperse particles at the
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Figure 4. Simulated structure of the Langmuir film consisting of averagely 44 nm diameter Sto¨ber silica particles at 10 mN/m surface pressure. The color-coding corresponds to the number of neighbors: 3, dark blue; 4, green; 5, red; 6, white; 7, yellow.
Figure 5. Radial pair-correlation function of the structure shown in Figure 4, where rcc is the distance between the centers of particles and rmean is the mean center-to-center distance between two neighboring particles.
same surface pressure. The coverage of the plane is 71.6% in Figure 4, whereas the monodisperse hexagonal system provides 74.6% coverage at 10 mN/m. This discrepancy is considerable, since the realistic system can reach 74.6% surface coverage only at about two times greater (≈20 mN/m) surface pressure. Not surprisingly, the bigger particles have more neighbors on average. Using the color codes of Figure 4, the yellow particles are from the greater diameter tail of the size distribution, and the green ones are among the smallest sized particles. The degree of order can be quantified by means of the paircorrelation function. Although some results presented in the next section indicate that the average distance between the particles can be slightly smaller in the direction of compression than perpendicular to it, it is sufficient to investigate the radial paircorrelation function because the decay length of the interaction is much smaller than the standard deviation of the particle diameters. According to the radial pair-correlation function shown in Figure 5, the positions of particles in the Langmuir film of 44 nm diameter Sto¨ber silica correlate up to six average particleparticle distances. The decay length of the pair-correlation function is a welldefined measure for the characteristic length of the order in the film. In noncrystalline condensed phases, far from phase transitions, the envelope of this function is often an exponential decay.23 In Figure 6 the lg(abs(g - 1)) transformation of the pair-correlation is presented, allowing us to determine the decay length. A straight line can be fitted to the local maxima of the transformed curve up to about six center-to-center distances, and beyond this point, the function behaves randomly. The length (L) after which the autocorrelation of the structure decreases (23) Leote de Carvalho, R. J. F.; Evans, R.; Hoyle, D. C.; Henderson, J. R. J. Phys. Condens. Matter. 1994, 6, 9275-9294.
Agod et al.
Figure 6. Transformed function of the pair-correlation shown in Figure 5 to determine the characteristic length of the order in the Langmuir film of 44 nm Sto¨ber silica particles.
Figure 7. Structure of a Langmuir film consisting of particles having 2.3% polydispersity at 10 mN/m surface pressure. The colors refer to the number of neighbors, and the coding is equivalent to that in Figure 4. The arrow is pointed at a vacancy, which disappears at 11 mN/m surface pressure.
with a decade can be derived from the slope of the linear fit. In the case of 44 nm Sto¨ber silica system, L ) 2.71 mean p-p distance, or about 131 nm. Effects of Polydispersity. In this subsection, we explore the relation between the broadness of the size distribution and the resulting order of the Langmuir film. The polydispersity is characterized by the ratio of the standard deviation (σ) and the mean of particle diameters (d). In a series of simulated experiments, we set the diameters to have normal distribution with 44 nm mean and varied the polydispersity between 0% and 15.8% (characteristic of 44 nm Sto¨ber silica). The interaction potential given by eq 5 was used between the particles. Our results indicate that for low polydispersity systems (σ/d < 0.07-0.08) domain structure is expected, i.e., the particles are situated at the liquid-fluid interface in nearly hexagonal arrays separated by line defects. In Figure 7 we present the structure of a Langmuir film consisting of particles having 2.3% polydispersity at 10 mN/m surface pressure. In a domain-structured film, the principal axes of the domains can intersect at an arbitrary angle (Figure 7), so the particle positions in different domains are not correlating, and this way the mean linear size of the domains can be estimated by the decay length of the pair-correlation function (L). According to the results of the simulation the ordering of Langmuir films, i.e., the size of hexagonal domains, is decreasing with increasing polydispersity. The decay lengths of the pair-correlation functions determined by the model are shown in Figure 8 as a function of polydispersity. In Figure 7 the color of a particle corresponds to the number of its direct neighbors. As can be seen, inside a domain most particles have six neighbors, whereas at the boundary this number
Modeling the Structure Formation of Langmuir Films
Figure 8. Decay length of pair-correlation (L) as a function of polydispersity (σ/ d). L is measured in terms of the average distance between the centers of two neighboring particles (rmean). The gray bar indicates the 7-8% polydispersity range.
Figure 9. Average number of neighbors as a function of polydispersity. The gray bar indicates the 7-8% polydispersity range.
is usually either five or four. It follows that the average number of neighbors (Nn) is a good measure in case of domain-structured films because, the smaller the domain size, the larger the ratio of particles situated at boundaries, thus the smaller Nn. In Figure 9 the average number of neighbors is presented as a function of polydispersity. Up to 7-8% polydispersity, where domainstructure formed, Nn is monotonously decreasing. However, for broader size distribution systems (σ/d > 0.07-0.08) there are no well-defined hexagonal domains and line defects separating them; these films are rather similar to the amorphous layer demonstrated in Figure 4. Beyond 7-8% polydispersity, the average number of neighbors is not a good measure of the structure any more; according to Figure 9, its value does not vary significantly. Contrary to the average number of neighbors, the standard deviation of number of neighbors (σNn) is a monotonous function of the polydispersity, so it can characterize not only domainstructured Langmuir films. To interpret the relation between σNn and the polydispersity demonstrated in Figure 10, it is advisable to treat low- and high-polydispersity systems separately. In the case of small polydispersity, the increase of the standard deviation of neighbors has similar reasons as the decrease of the average number of neighbors. The smaller the domains, the more particles have a coordination number of 4 or 5 instead of 6, and this way the greater the variance of the number of neighbors. However, in systems of high polydispersity (σ/d > 0.07-0.08), more and more particles can have seven or three neighbors simply because the big particles can be surrounded by small ones and vice versa, resulting in the further increase of σNn (see Figures 4 and 10). Therefore, the increase in the standard deviation of number of neighbors has different roots in low- and high-polydispersity systems, and according to Figure 10, the increase gets less steep in the transition region (σ/d ≈ 0.07-0.08). In electron microscopy images of particulate Langmuir films, one can often see vacancies in the middle of domains. According
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Figure 10. Standard deviation of number of neighbors as a function of polydispersity. The gray bar indicates the 7-8% polydispersity range.
Figure 11. Global orientational order parameter as a function of polydispersity. The gray bar indicates the 7-8% polydispersity range.
to Figure 7 the simulation can reproduce this type of lattice defects. The results of the modeling indicate that the probability of vacancy formation is decreasing as the size distribution becomes broader. In a perfectly monodisperse system, there is one vacancy for about 500 particles, and beyond 7-8% polydispersity we have not observed any vacancies. Our explanation for this trend is that the more different-sized particles build up the film, the less stable these point defects are against compression. In the case of a broader size distribution, the break-up of the ring around a vacancy is more probable because a small particle can just fall into the empty area. In Figure 7, the arrow points at a vacancy that exists at 10 mN/m but disappears at 11 mN/m surface pressure. The deviation of a structure from perfect hexagonal order can be characterized by the global orientational order parameter introduced by Zangi (eq 6).24,25 n
1 N 1 i i6Θij |Φ6| ) | e | N i)1 ni j)1
∑ ∑
(6)
Here N is the total number of particles in the layer and ni is the number of neighbors of particle i. The index j runs through the direct neighbors of the ith particle, and Θij is the angle between an arbitrary but fixed straight line and the line determined by the centers of particle i and j. The definition of the orientational order parameter (eq 6) suggests that its value is close to 1 if the ij directions have an angle of integral multiple of 60° between each other, i.e., the order is close to hexagonal. This parameter is very sensitive to any deviation from hexagonal order, and its value is close to zero in case of a random structure. In Figure 11 the dependence of the orientational order parameter on the polydispersity is demonstrated. Most of the parameters we used to describe the structure of Langmuir films had a change in their trend at about 7-8% (24) Zangi, R.; Rice, S. A. Phys. ReV. E 1998, 58, 7529-7544. (25) Sun, J.; Stirner, T. Phys. ReV. E 2003, 67, 51107.
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Agod et al.
Figure 12. Simulated anisotropy of surface pressure (Πxx/Πyy) as a function of surface area per particle for a 44 nm, monodisperse system and for 44 nm Sto¨ber silica particles. The particulate layers were compressed in the direction of the y axis.
polydispersity, but most significantly, the orientational order parameter varied. This indicates that the domain-structure disappears when the local triangular order cannot be sustained any more because of the variation of particle sizes. Anisotropy of the Surface Pressure. We characterized the anisotropy of the surface pressure by the ratio of the diagonal elements of the pressure tensor (Πxx/Πyy). The surface pressure tensor can be derived from the virial theorem just like the scalar surface pressure (eq 7):19
Πab )
1 DA
N
〈
1
miVi,aVi,b〉 + 〈∑ Fij,arij,b〉 ∑ DA i)1
(7)
(i,j)
The symbols we have not defined after eq 1 are Fij,a, which is the a component of the force between particles i and j, rij,b, which is the b component of the vector connecting the centers of particles i and j, and Vi,a, which stands for the a component of the velocity of particle i. In Figure 12 the anisotropy of the pressure tensor is shown as a function of surface area per one particle (A1) for monodisperse particles and for a system with 15.8% polydispersity. (The range of A1 starts from the collapse area and ends at the surface area per particle where the surface pressure reached 1 mN/m.) It is important to note that the fluctuation of Πxx/Πyy in Figure 12 is solely because of the finite size of the simulated system; therefore, in real experiments we should not be able to measure it by a Wilhelmy plate. In most cases the finite size effects in the simulations are undesirable side effects, but here it reveals an important mechanism of the structure formation. A compact monolayer cannot easily restructure during the uniaxial compression. A Langmuir film does not have to be cohesive to get stiff and hard to rearrange, because the steep decay of p-p potential does not let the interparticle distances to vary considerably. (The range of interaction is much smaller than the mean diameter of particles.) Therefore, in the course of the necessary restructuring due to the unidirectional compression, the local order is preserved. The system follows the constraint dictated by the moving barrier of the film balance by collective motions of the particles; thus, as opposed to individual particles, whole domains are the basic units of restructuring. We presented an example for these collective motions in Figure 13 for the simulated film of 44 nm Sto¨ber silica system. The lines ending in the center of particles show the directions where the particles arrived from. For the sake of better visibility, in the right-hand side of the figure only the displacement field is shown. According to Figure 13 the particles shift not only in the direction of compression but also perpendicular to it. If it were
Figure 13. Simulated restructuring of 44 nm diameter Sto¨ber silica particles. The lines ending in the center of particles show the directions from which the particles arrived. The direction of the compression is vertical in the figure.
not like this, Πxx would remain zero. However, a locally stiff structure not easily and in most cases not continuously adapts to the steadily changing external constraints (i.e., to the decreasing surface area due to the moving barrier), which results in the local fluctuation of the ratio of surface pressure measured in x and y directions. In a system of readily restructuring particles (ideal liquid) Πxx/Πyy would be constantly 1, but in our particulate system with short-range p-p interactions this ratio is fluctuating around a value which is less than 1. Increasing the size of the system to infinity this fluctuation would disappear, and the ratio of the surface pressures in the perpendicular directions would be constant and its value would depend on the polydispersity. According to Figure 12 in the polydisperse system of 44 nm Sto¨ber silica particles the anisotropy is fluctuating around 0.9 with relatively low deviation ((0.05). In the Langmuir film of monodisperse particles, the anisotropy is greater, the ratio of the diagonal elements of the pressure tensor is roughly 0.8, and the amplitude of the fluctuation is more significant (≈0.1). Thus, we can conclude that the restructuring of a polydisperse system is a smoother process and its structure is less stiff. We showed in the previous section that the characteristic length of the order, i.e., the decay length of the pair-correlation function decreases with increasing polydispersity. In this context it means that during restructuring smaller units of the film move relative to each other, i.e., the system adapts to the external constraints more easily. We found that the anisotropy of the surface pressure changes abruptly when relatively big parts of the monolayer shift suddenly. In these cases the stress gets more homogeneous and the angular dependence of the surface pressure decreases. In the monodisperse system the local fluctuation of the anisotropy of the surface pressure is more significant, which indicates again that bigger units move during the restructuring of a more ordered film.
Conclusions We developed a molecular dynamics simulation to model the structure formation of partially wettable, spherical nanoparticles in film-balance experiments. By means of the simulation we proved that at a given surface coverage a real particulate film has greater surface pressure than an idealistic monodisperse, hexagonally packed system due to the uniaxial compression of real films and the polydispersity of nanoparticles. Consequently, the usual assumption in the determination of interparticle potentials from the surface pressure vs area isotherms (i.e., monodisperse particles in perfect hexagonal order) leads to a systematic overestimation of the characteristic length of the interaction. On the basis of the results of the simulation, we proposed a correction method to improve the traditional determination of the interparticle potentials.
Modeling the Structure Formation of Langmuir Films
The correlation between the broadness of size distribution and several structural parameters has been explored. We found that domain-structured films can form only below 7-8% polydispersity; beyond this limit the particulate films have rather amorphous structure. The domain size, as well as the average number of neighbors, is decreasing with increasing polydispersity. Without domain-structure the average number of neighbors cannot characterize the order, since its value does not vary significantly when the polydispersity changes. However, the standard deviation of number of neighbors is a monotonously increasing function of polydispersity; thus, it can be a good measure of the order in Langmuir films. The decay length of the pair-correlation function in a monodisperse system was about 15 average particle centerto-center distances, and it decreased monotonously to about 2.7 center-to-center distances after reaching 15.8% polydispersity (the value which is characteristic to 44 nm average diameter Sto¨ber silica particles.) The global orientational order parameter significantly decreased when crossing the 7-8% polydispersity limit, indicating that the domain-structure disappears because even the local triangular order cannot be sustained any more. The particulate Langmuir films are continuously restructuring in the course of the unidirectional compression. According to the
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results of the computer simulation, the adjacent particles usually shift in the same direction; therefore, the local structure is mostly preserved. Units consisting of numerous particles move collectively to follow the constraint dictated by the barrier. Because of the uniaxial compression and the stiffness of the particulate layer, the surface pressure is not a scalar field, its value depends on the direction angle. In systems of smaller polydispersity, bigger units move in the course of restructuring; thus, the ratio of diagonal elements of the local pressure tensor fluctuates more during compression. We found that the narrower the size distribution the more significant the difference between the values of surface pressure parallel and perpendicular to the barrier. Acknowledgment. This work was supported by the Hungarian Scientific Research Fund (OTKA T049156) and was accomplished in the scope of a specific support action of EU FP 6 (Hungarian Network of Excellent Centers on Nanosciences, HUNN) and of an Austrian-Hungarian cooperation program (Te´T A-3/04). LA063481U
