Article pubs.acs.org/Langmuir
Structural Ordering of Self-Assembled Clusters with Competing Interactions: Transition from Faceted to Spherical Clusters J. E. Galván-Moya,* K. Nelissen, and F. M. Peeters Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium ABSTRACT: The self-assembly of nanoparticles into clusters and the effect of the different parameters of the competing interaction potential on it are investigated. For a small number of particles, the structural organization of the clusters is almost unaffected by the attractive part of the potential, and for an intermediate number of particles the configuration strongly depends on the strength of it. The cluster size is controlled by the range of the interaction potential, and the structural arrangement is guided by the strength of the potential: i.e., the self-assembled cluster transforms from a faceted configuration at low strength to a spherical shell-like structure at high strength. Nonmonotonic behavior of the cluster size is found by increasing the interaction range. An approximate analytical expression is obtained that predicts the smallest cluster for a specific set of potential parameters. A Mendeleev-like table is constructed for different values of the strength and range of the attractive part of the potential in order to understand the structural ordering of the ground-state configuration of the selfassembled clusters.
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INTRODUCTION The self-assembly process in a system of individual entities is of interest in all areas of nature because of its spontaneity, its specific rules of grouping, and particularly, in some cases, its symmetry. When these entities are nanosize particles, one can address this grouping process in multiple ways. In nanotechnology, by definition,1 a self-assembled system is formed as a direct consequence of competing molecular interactions, which usually are modeled by a pairwise potential. Because of technological advances, the experimental observation of self-assembled nanoparticles has been made possible. These self-assembled systems, which from now on we will call clusters, have been observed in both two- (2D)2−4 and three-dimensional (3D)4,5 systems. In the last few years, the number of studies on these self-assembly processes has increased,5−12 indicating that this is a very promising area of study with a wide range of applications in many fields.13−16 Two-dimensional self-assembled systems have been investigated in the past using a competing interaction for the pairwise potential17−19 and considering a quadratic confinement potential in order to stabilize the system formation in a closed region. The self-organization and the pattern formation were also analyzed in 2D by applying different methods.20,21 Recently, experiments have shown that the interparticle interaction can be modeled by a pairwise competing potential in the case of self-assembled clusters of metallic nanoparticles,5,6,11 quantum dots at the air−water interface,2 and colloidal clusters by depletion attraction.22 These results show the importance of the competing interaction in the selfassembly process. © XXXX American Chemical Society
For 3D systems, theoretical studies have covered a wide range of different functional forms for the competing interaction, from a pure Lennard-Jones5,23 and Morse24,25 potential to a mixing between the Coulomb26 and Yukawa27 potential. However, in most of those studies, the analysis was restricted to the investigation of the symmetry of the formed structures, and in some cases, an external confinement was introduced in order to favor cluster formation. Recent techniques of 3D tomographic reconstruction of the experimentally observed self-assemblies28,29 allow one to recover the number of particles per assembly and their position, resulting in a complete structural description of the system. In 2012, a theoretical model was presented to analyze the structural organization of self-assembled gold nanoclusters that were found experimentally.11 This model assumed that the inclusion of a hydrophobic interaction is essential to understanding the aggregation mechanism, in addition to van der Waals interactions.30,31 Although the model proposed in ref 11 may very well predict interparticle distances in the cluster, it turned out that the experimentally found 3D configurations of assemblies could not be obtained. The reason could be traced back to the fact that the van der Waals interaction considered is correctly described only in the limit of close approach and overestimates the strength of the long-range interaction. To overcome this problem, we recently proposed a new model based on a pairwise interaction between particles that enables us to tune Received: October 31, 2014 Revised: December 22, 2014
A
DOI: 10.1021/la504249e Langmuir XXXX, XXX, XXX−XXX
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Langmuir the strength and the range of the attraction.32 A comparison with recent experiments showed the high accuracy and the predictive power for the structure of clusters of gold nanoparticles. In this work we provide a thorough interpretation of the process of cluster formation. We show the usefulness of the new model for the analysis of different kinds of nanoparticles, irrespective of the composing material. We determine the arrangement of the nanoparticles inside the cluster and determine the range of parameter values of the potential for which faceted and spherical clusters will be found. The results are summarized in Mendeleev-like tables.
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MODEL SYSTEM In this work, we consider a system of N particles with electric charge q moving in a medium with a dielectric constant ε, interacting through a pairwise interaction in the absence of external forces, allowing the formation of self-assembled clusters. The total energy of the system is given by N
E=
N
∑ ∑ Vpair(rij) (1)
i=1 j>i
where rij represents the relative distance between the ith and jth particles. The pairwise interaction potential has the form Vpair(r) = Urep(r) − Uatt(r) and consists of a repulsive (Urep(r)) and an attractive (Uatt(r)) term. In general, the interaction considered in this work is given by Vpair(r ) =
q2Rm − 1e−αr q2Rn − 1e−βr − B̃ m εr εr n
Figure 1. (a) Landscape of the pairwise interaction potential defined by eq 4, where the minimum energy Emin is located at a distance r0. (b) Contour plot of Emin as a function of parameters B and β. Isolines are depicted for several values of r0, and the dashed gray curve follows the minimum in the stability lines drawn in red.
and the minimization of r0 with respect to β, i.e., (dr0/dβ) = 0, gives us the relation
(2)
where parameters α(β) and m(n) control the range of the repulsion (attraction) between particles and B̃ modulates the strength of the attraction potential. Distance R is an arbitrary length parameter introduced to guarantee the right units. In dimensionless form, the pairwise interaction becomes Vpair(r ) =
e −r e − βr − B rm rn
B = β e(1 − β)/ β
In Figure 1b we show the contour plot of Emin as a function of B and β, where some isolines of r0 (eq 5) are drawn with red curves for several values, as indicated. The dashed white curve is the path followed by the minimum in r0, which is given by eq 6. From that figure one can see that r0 increases with increasing B whereas the effect of β is not monotonic, allowing the definition of the (B, β) position of the smallest interparticle distance, i.e., the lowest value of r0. Several studies addressed the problem of the self-assembly of particles by using a competing interaction,17−19,33 but in most cases an external confinement was introduced in order to obtain stable structures. The advantage of the present pairwise interaction is that, in the considered region (B < 1 and β < 1), there is a unique global minimum as shown in Figure 1 together with a long-range attraction. The latter allows for the selfassembly process of interacting particles without any external confinement potential, as was proven recently for the case of the self-assembly of gold nanoclusters.32 Using Monte Carlo simulations (MC) supplemented with the Newton optimization method,34 we calculate the groundstate configuration of the system. As a first result, irrespective of the parameter values, we notice that the model is translationally and rotationally invariant. Highly symmetric configurations have been found in accordance with the recently reported configurations for Au nanoclusters.32
(3) 2 m−1
/εr0m,
The energy of eq 3 is normalized by E0 = q R and all distances are scaled in units of r0 = 1/α. β is expressed in units of 1/r0, and the strength parameter is redefined as B = B̃ (r0/ R)m−n. As a simple case for the pairwise potential, we consider a model where the range of the repulsion and attraction terms are comparable (m = n = 0), allowing the attractive interaction to become dominant for large interparticle distances (β < 1). In this approximation, eq 3 becomes Vpair(r ) = e−r − Be−βr
(4)
This model can be understood as a generalization of the Morse potential24,25 as well as an extension of the phenomenological pairwise interaction proposed in refs 2 and 20, which has been used successfully to describe the structural formation of 2D selforganized structures in colloidal systems. A schematic representation of the interaction potential is shown in Figure 1a, indicating the interparticle distance (r0) with its corresponding minimum energy (Emin). The location of r0 can be calculated analytically as follows
r0 =
ln(Bβ) β−1
(6)
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RESULTS AND DISCUSSION In the present section we present a systematic study of the structural properties of the clusters. We limit ourselves to those clusters that are ground-state configurations. This analysis is
(5) B
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Figure 2. Some typical configurations for clusters with a small number of particles, as indicated in each image. The yellow bar represents a symmetry axis of the system, and each drawn plane highlights a group of particles equidistantly separated from the axis. The different colors of the particles indicate different radial distances from the center of mass.
the ground-state configuration consists of shell-like structures, following the process described in ref 32. As the number of particles increases, the radius of the shells increases, leading to the creation of a void at the center of the cluster. However, we found that the formed ground-state structure depends strongly on parameters B and β. As expected, the number of stable states increases with N, generating a very rich phase diagram for the ground state of systems with intermediate and large numbers of particles, as was presented in ref 32 for N = 59. Intermediate Number of Particles. As a particular case, we analyze the structural properties of the ground state for a system with N = 38 particles. The phase diagram of this system is shown in Figure 3a, where the regions of the different phases are plotted as a function of parameters B and β. Four different phases are found as the ground-state configuration, and in all cases, the phase transition is first order. The white region above phase 1 represents structures that are not spherical-like, which is a consequence of the short range of the attractive interaction. In Figure 3b we show the lowest nonzero eigenfrequencies of the normal modes as a function of parameter B for β = 0.6. Because the model is translationally and rotationally invariant, six of the normal eigenfrequencies are zero. The vertical dashed lines indicate the position of the first-order phase transitions. Although the ground-state configuration for this system is found to be arranged in shell-like structures, these configurations are not composed of spherical shells of particles. For weak and short-range attractive interactions (white region above phase 1 in Figure 3), the ground-state configuration is mainly guided by local interactions, resulting in faceted structures such as the ones found earlier for the LennardJones interparticle interaction.23,35−37 Therefore, the analysis of the structural transition between the different phases results in a very useful tool for understanding the effect of the different parameters of the competing interaction on the ground-state configuration. In Figure 4 we plot the distance between the center of mass of the cluster and each particle in the cluster, with different colors serving as a reference for the reader indicating the group of particles located at the same radial distance. The 3D configuration is drawn in the inset of each figure, showing the particles in color to help the reader to understand the structural organization in each case. It is important to bear in mind that the phases are numbered according to the strength of the attractive interaction in the region where they are found, from short range (phase 1) to long range (phase 4). From Figure 4 one can see that, for phase 1, even when the two different shells can be clearly identified, the structural
Figure 3. (a) Phase diagram of the ground-state configuration of a system with N = 38 as a function of parameters B and β. (b) Lowest nonzero eigenfrequencies of the ground state as a function of B for β = 0.6.
done for three different regimes, according to the number of particles in the cluster. The influence of the strength and the range of the interparticle interaction on the self-assembled clusters is investigated. Finally, a detailed study of the structural properties of spherical clusters, found as ground state for a limited region of the values of the theoretical parameters, is performed. Small Number of Particles. For a small number of particles (N), the arrangement of the ground-state configuration of the clusters is given by different polyhedral structures with the particles located at each vertex, as shown in Figure 2, where the highlighted planes and the thin lines are drawn as a guide for the eye. For N ≤ 10, the configurations are robust when changing B and β within a range of values. For N > 10, C
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Figure 5. Phase diagram of the ground state for a system with N = 125 particles as a function of parameters B and β. The configurations of the different numbered phases are presented in Table 1.
Table 1. Shell-like Configuration of the Different Phases Found as the Ground State for a System of N = 125
Figure 4. Radial distance of each particle from the center of mass of the cluster for the different phases found as a ground-state configuration of a system with N = 38 particles. The different colors indicate particles at the same radial distance. The inset in each plot shows a view of the 3D configuration of the system.
phase
configuration
phase
1 2 3 4 5 6 7
(1, 14, 43, 67) (1, 14, 44, 66) (1, 13, 41, 70) (1, 12, 40, 72) (10, 39, 76) (10, 38, 77) (10, 37, 78)
8 9 10 11 12 13 14
configuration (9, (9, (9, (9, (8, (8, (7,
37, 36, 35, 34, 34, 32, 32,
79) 80) 81) 82) 83) 85) 86)
phase 15 16 17 18 19 20 21
configuration (6, (6, (6, (6, (5, (5, (4,
32, 30, 29, 28, 28, 27, 27,
87) 89) 90) 91) 92) 93) 94)
Figure 6. Size of the cluster, with N = 125, as a function of range parameter β for different values of B; the dashed blue line follows the minimum in the curves. The inset shows the (β, B) position of the minimum cluster diameter (dashed blue curve) and the minimum interparticle distance of the pairwise interaction (r0) from eq 6 (solid green curve).
configuration holds some similarities with a faceted structure (white region in Figure 3). Indeed, each shell can be subdivided into smaller structures, as can be seen for the inner shell where the nine particles are arranged in three stacked planes that form a triangular arrangement of particles (similar to the configuration for N = 9 shown in Figure 2). Similar behavior occurs for the outer shell, where the number of stacked planes increases; however, the triangular configuration is still present in each plane. By following the phases in ascending order, i.e., increasing the strength of the attractive interaction, for large β values one can observe that particles located in the same shell tend to form a spherical structure; e.g., in phases 3 and 4 the formed
structure is almost spherically symmetric, meaning that groups of particles (marked with the same color in Figure 4) are located at similar radial distances. This behavior shows that the formation of spherical clusters is a direct consequence of the short-range attraction of the pairwise interaction in the system, and it is mainly guided by the strength of the attraction. Large Number of Particles. In this section we analyze the ground-state configuration for a system of N = 125. The phase diagram of the ground state as a function of B and β is presented in Figure 5, where the different phases are numbered and their respective configurations are given in Table 1. D
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Langmuir Table 2. Mendeleev-like Table for Different Sets of Parameters B and βa β = 0.2
β = 0.5
β = 0.2
β = 0.8
β = 0.5
β = 0.8
N
B = 0.8
B = 0.9
B = 0.7
B = 0.8
B = 0.9
B = 0.7
B = 0.9
N
B = 0.8
B = 0.9
B = 0.7
B = 0.8
B = 0.9
B = 0.7
B = 0.9
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
(3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (1, 12) (1, 13) (1, 14) (1, 15) (1, 16) (1, 17) (1, 18) (2, 18) (2, 19) (2, 20) (2, 21) (2, 22) (3, 22) (4, 22) (3, 24) (4, 24) (4, 25) (4, 26) (4, 27) (5, 27) (6, 27) (6, 28) (6, 29) (6, 30) (6, 31) (6, 32) (7, 32)
(3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (1, 12) (1, 13) (1, 14) (1, 15) (1, 16) (1, 17) (1, 18) (1, 19) (1, 20) (2, 20) (2, 21) (2, 22) (3, 22) (2, 24) (3, 24) (4, 24) (4, 25) (4, 26) (4, 27) (4, 28) (4, 29) (4, 30) (5, 30) (6, 30) (6, 31) (6, 32) (6, 33)
(3) (4) (5) (6) (7) (8) (9) (10) (1,10) (12) (1, 12) (1, 13) (1, 14) (1, 15) (1, 16) (1, 17) (1, 18) (2, 18) (2, 19) (2, 20) (2, 21) (3, 21) (3, 22) (4, 22) (4, 23) (4, 24) (4, 25) (4, 26) (4, 27) (5, 27) (6, 27) (6, 28) (6, 29) (6, 30) (6, 31) (6, 32) (7, 32)
(3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (1, 12) (1, 13) (1, 14) (1, 15) (1, 16) (1, 17) (1, 18) (2, 18) (2, 19) (2, 20) (2, 21) (2, 22) (3, 22) (3, 23) (3, 24) (4, 24) (4, 25) (4, 26) (4, 27) (4, 28) (5, 28) (6, 28) (6, 29) (6, 30) (6, 31) (6, 32) (7, 32)
(3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (1, 12) (1, 13) (1, 14) (1, 15) (1, 16) (1, 17) (1, 18) (1, 19) (1, 20) (2, 20) (2, 21) (2, 22) (2, 23) (2, 24) (3, 24) (3, 25) (3, 26) (3, 27) (4, 27) (4, 28) (4, 29) (4, 30) (4, 31) (4, 32) (5, 32) (6, 32) (6, 33)
(3) (4) (5) (6) (7) (8) (9) (1, 9) (1,10) (1,11) (1, 12) (1, 13) (1, 14) (1, 15) (1, 16) (2, 16) (2, 17) (2, 18) (3, 18) (4, 18) (4, 19) (4, 20) (4, 21) (4, 22) (6, 21) (6, 22) (6, 23) (6, 24) (6, 25) (6, 26) (7, 26) (7, 27) (8, 27) (9, 27) (9, 28) (9, 29) (9, 30)
(3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (1, 12) (1, 13) (1, 14) (1, 15) (1, 16) (1, 17) (1, 18) (2, 18) (2, 19) (2, 20) (2, 21) (2, 22) (3, 22) (4, 22) (3, 24) (4, 24) (4, 25) (4, 26) (4, 27) (5, 27) (6, 27) (6, 28) (6, 29) (6, 30) (6, 31) (6, 32) (7, 32)
40 41 42 43
(8, (8, (8, (8,
(6, (6, (6, (7,
(8, (8, (8, (9,
(6, (7, (7, (8,
(6, (6, (6, (6,
(8, (8, (8, (8,
44
(8, 36)
(7, 37)
(9, 35)
(8, 36)
(7, 37)
45
(9, 36)
(8, 37)
(9, 36)
(8, 37)
(7, 38)
46
(9, 37)
(8, 38)
(9, 37)
(8, 38)
(7, 39)
47
(9, 38)
(8, 39)
(10, 37)
(9, 38)
(8, 39)
48
(9, 39)
(9, 39)
(10, 38)
(9, 39)
(8, 40)
49
(10, 39) (10, 40) (10, 41) (10, 42) (12, 41) (12, 42) (12, 43) (12, 44) (1, 12, 44) (1, 12, 45) (1, 12, 46) (1, 13, 46)
(9, 40)
(10, 39)
(9, 40)
(8, 41)
(9, 41)
(10, 40)
(9, 41)
(9, 41)
(9, 42)
(12, 39)
(10, 41) (12, 40) (10, 42) (12, 41) (10, 43) (12, 42) (10, 44) (1, 12, (11, 42) 44) (1, 12, (12, 43) 44) (1, 12, (12, 44) 45) (1, 13, (12, 44) 46) (1, 14, (1, 12, 44) 46) (1, 14, (12, 45) 48)
(9, 42)
(10, 30) (10, 31) (10, 32) (1, 12, 30) (1, 12, 31) (1, 12, 32) (1, 13, 32) (1, 12, 34) (1, 13, 34) (1, 14, 34) (1, 14, 35) (1, 14, 36) (1, 15, 36) (1, 15, 37) (1, 15, 38) (1, 16, 38) (1, 16, 39) (1, 16, 40) (1, 17, 40) (1, 17, 41) (2, 18, 40)
50 51 52 53 54 55 56 57 58 59 60 a
32) 33) 34) 35)
34) 35) 36) 36)
(10, 42) (10, 43) (10, 44) (10, 45) (10, 46) (12, 45) (12, 46) (12, 47) (12, 48)
32) 33) 34) 34)
34) 34) 35) 35)
34) 35) 36) 37)
(9, 43) (9, 44) (10, 44) (10, 45) (10, 46) (10, 47) (10, 48) (11, 48) (12, 48)
32) 33) 34) 35)
(8, 36) (9, 36) (9, 37) (9, 38) (9, 39) (10, 39) (10, 40) (10, 41) (10, 42) (12, 41) (12, 42) (12, 43) (12, 44) (12, 45) (1, 12, 45) (1, 12, 46) (1, 13, 46)
The clusters highlighted in bold are magic number configurations.
when β increases, one can observe that, for a fixed value of B, the cluster size decreases for small values of β until it reaches its minimum at βmin, from where the cluster size starts to increase monotonically. This βmin indicates the optimal β value for finding the smallest spherical-like configuration. The dashed blue curve shows the relation between β min and its corresponding cluster diameter. The inset in Figure 6 shows the (B, β) positions of the smallest cluster configuration by the dashed blue curve and of the minimum in the pairwise interparticle distance (r0) by the solid green curve; the latter (B, β) positions are obtained from eq 6. As an interesting result, one can see that these curves are very close to each other, indicating a similarity in the behavior of both features. However, both curves are closer when the attractive interaction is short range, i.e., large values of β, with high strength. In this case the effective interaction of each particle is dominated by its nearest neighbors; consequently, the whole cluster behaves similarly to a two-body system. Oppositely, for long-range attraction the smallest cluster is formed for small values of B. In this case, each particle is influenced by the interaction with all the rest, and it diminishes
The phase diagram shows sequential behavior. In general, the packing fraction of the system increases with decreasing B allowing the formation of spherical structures for larger values of it.32 Similar to the situation for N = 38, all phase transitions are first order. The configuration in the region above phase 1 of Figure 5 is a nonspherical shell-like structure. A complete list of the ground-state configurations for this system is presented in Table 1, in which the ordering of the numbered phases has been chosen as the order of appearance of the configuration with increasing B for a fixed range of the interparticle interaction, i.e., β = constant. From Table 1 one can see that the number of shells in the ground-state configuration is reduced with increasing B, resulting in more spherical-like configurations. This formation process is guided by the transfer of particles located in the inner shells toward the outer shells. In Figure 6 we present the size of the cluster as a function of β for different values of B. This figure shows that even when the cluster size is monotonically reduced by increasing the strength of the interaction (B), the effect of the interaction range (β) is more complex, allowing one to tune the minimum cluster diameter. When the range of the interaction decreases, i.e., E
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This finding reveals a very useful feature of these systems: the smallest spherical-like configuration of a cluster can be understood as the result of the minimization of the pairwise interparticle distance, and this behavior is accentuated when the range of the pairwise interaction is short. When the pairwise interaction is known analytically, as in our case, the parameters of the interaction potential (B and β) can be analytically calculated, i.e., without any previous simulation. Spherical-like Structures. To perform a systematic study of spherical-like structures, we will analyze the effect of the screening parameter (β), i.e., the range of the attraction, on the ground-state configuration for large values of B. In Table 2 we present a Mendeleev-like table for the ground-state configuration up to N = 60 for different sets of parameters B and β. The bold results in Table 2 indicate the most stable configurations. Those configurations are very symmetric and almost invariant when changing the parameters of the pairwise interaction. They are known as magic particle numbers. We can also summarize the results of Table 2 in Figure 7, where we plot the number of particles per shell as a function of the total number of particles of the system for (a) B = 0.7 and (b) β = 0.5. From Figure 7a one can see that reducing the range of the attraction potential, i.e., increasing β, leads to the emergence of a new shell at a small value of N. Conversely, Figure 7b shows that by increasing B the emergence of a new shell appears for a larger N value. This behavior evidences the large impact of the range of attraction on the process of shell formation, thus for larger β values the size of the cluster diminishes and the interaction between particles in the different shells increases. In Figure 7c we plot the cluster diameter of the ground-state configuration as a function of the number of particles per cluster for different values of B and β as indicated. From this figure one can see that, even when the value of the screening parameter plays an important role in the formation of the structure, the size of the system is predominantly defined by the strength of the attractive interaction (B). Then, larger clusters can be formed just by reducing the strength of the attractive interaction; however, Figure 7c shows that this effect is attenuated when the range of the attractive interaction is large (small values of β). Our simulations for a large number of particles show that the organization of the particles in the inner shells follows the sequence as given in Table 2, irrespective of the number of particles. In Figure 8 we show a 3D view of the ground-state configuration for systems with N = 250 and 500, with shell-like structure given by (6, 32, 68, 144) and (6, 30, 73, 126, 265), respectively, for B = 0.65 and β = 0.5. Notice that the inner shell of the clusters with N = 250 and 500 particles coincides with those obtained for N = 38 and 36, respectively (Table 2). These results allow us to conclude that the most important features of the structure of the cluster are already present in clusters with an intermediate number of particles.
Figure 7. Number of particles per shell as a function of the total number of particles of the cluster for (a) B = 0.7 and (b) β = 0.5. (c) Cluster diameter of the ground-state configuration as a function of the total number of particles for different set of parameters (B, β) as indicated to the right of the figure.
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CONCLUSIONS We investigated the self-assembly of nanoparticles interacting via a competing pairwise potential. The ground-state configuration was found numerically, and the effects of the parameters characterizing the interparticle interaction on the self-assembly process were investigated in detail. Our simulations showed that the number of particles per shell can be controlled by the range of the attractive interaction. However, the strength of the attractive potential is primarily
Figure 8. Ground-state configuration for systems containing a large number of particles for parameters B = 0.65 and β = 0.5: (left) N = 250 (6, 32, 68, 144) and (right) N = 500 (6, 30, 73, 126, 265).
the similarity between the behavior of the smallest cluster and that of the two-particle system. F
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Langmuir
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responsible for the size of the cluster and the organization of the particles over the different shells. Our analysis revealed that the appearance of spherical-like configurations is a direct consequence of the short-range attractive part of the pairwise interaction, and this process is mainly guided by the increasing strength of the attraction, whereas for low strength faceted-like structures are found. In the case of spherical-like clusters we found that, for a fixed value of the strength of the attractive interaction, the smallest cluster is realized for a particular value of the attraction range, for which an approximate analytic expression could be obtained from the pairwise interaction potential. This allows us to predict the set of potential parameters that will result in the smallest clusters without any previous simulation. The main results of our paper are summarized in Mendeleev-like tables. From these results one can conclude that, even when the distribution of the particles per shell varies depending on the values of the interaction parameters, this variation occurs in a systematic way, which allows us to control the number of particles per shell and to control the creation of tailor-made nanoclusters.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Flemish Science Foundation (FWO-Vl) and the Methusalem programme of the Flemish government. Computational resources were provided by the HPC infrastructure of the University of Antwerp (CalcUA), a division of the Flemish Supercomputer Center (VSC).
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DOI: 10.1021/la504249e Langmuir XXXX, XXX, XXX−XXX
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DOI: 10.1021/la504249e Langmuir XXXX, XXX, XXX−XXX
