16071
2007, 111, 16071-16075 Published on Web 10/09/2007
Controlling the Shape of Nanocrystals Sougata Pal, Biplab Goswami, and Pranab Sarkar* Department of Chemistry, VisVa-Bharati UniVersity, Santiniketan -731235, India ReceiVed: June 26, 2007; In Final Form: August 30, 2007
A theoretical framework is used to study the mechanism and energetics for the formation of nanocrystals having a particular shape. It is shown that, when nanocrystals are formed by a diffusion mechanism followed by nucleation, the presence of diffusion couple differing in diffusion rates plays an important role in determining the shape of the nanocrystal. Thus, by proper choice of diffusion partner, one can control the nature of building blocks and hence the shape of the nanocrystal.
Nanomaterials have represented a fundamental component of nanoscience and nanotechnology in the 21st century.1-6 Research works in nanomaterials face many challenges in synthesis, property characterization, and device fabrication. The small size and large diversity in the shapes of nanostructures are particularly attractive for exploring many unique and novel properties. The process for controlling the shape of inorganic nanocrystals is important for understanding basic size- and shape-dependent scaling laws and may be useful in a wide range of applications. There has been considerable progress along this line of investigation,7-11 with special emphasis on the growth and synthesis of nanocrystals of a particular shape. For example, one of the popular methods of growing spherical CdSe nanocrystals is to inject precursors that undergo pyrolysis into a surfactant (90% trioctylphosphine oxide, TOPO) at temperatures above 300 °C.7 Peng et al.8 found that anisotropic CdSe crystals could be grown using a combination of two pure surfactants: TOPO (99%) and an alkyl phosphonic acid (hexyl phosphonic acid). The same group also showed that a fine control of nanocrystal shape could be obtained by varying the ratio in which these surfactants were mixed.9,10 In recent years, studies on the synthesis and properties of hollow nanocrystals have become very popular since hollow nanocrystals offer possibilities in material design for applications in catalysis, nanoelectronics, nano-optics, and drug delivery systems, and also form the building blocks for lightweight structural materials.12-15 The ability to manipulate the structure and morphology of hollow materials on a nanometer scale would enable us to have control over a local chemical environment to a greater extent. Recently, Yin et al.11 synthesized hollow nanocrystals through a mechanism analogous to the Kirkendall effect.16 They also showed that the formation of pores occurs as a result of the difference in diffusion rates between the two components in a diffusion couple. For cobalt nanocrystals, they found that their reaction in solution with oxygen accompanied by either sulfur or selenium leads to the formation of hollow nanocrystals of the resulting oxide and chalcogenides. This process provides a general route for the synthesis of hollow nanocrystals of a large number of compounds. These authors have demonstrated that * Corresponding author. E-mail:
[email protected].
10.1021/jp074950j CCC: $37.00
nanoscale pores can be developed inside nanocrystals with a mechanism analogous to the void formation in the Kirkendall effect, in which mutual diffusion rates of the components in a diffusion couple differ by a considerable amount. The aim of the present communication is to have a theoretical understanding for the experimental observation of Yin et al.,11 that is, the mechanism of the formation of hollow nanocrystals when the difference of diffusion rates in a diffusion couple is of considerable amount. In other words, we would like to theoretically investigate the role of a diffusion couple in determining the shape of a particular nanocrystal when the nanocrystals are formed through a mechanism of diffusion followed by nucleation. Considering the ZnS nanocrystal as a prototypical example,17 one can think that its formation takes place through the following steps: (i) diffusion of sulfur into a suspension of Zn nanocrystals, (ii) formation of structural units or building blocks, and, finally, (iii) nucleation or self-assembly of these building blocks at some nucleation center. In this work we shall focus our attention on the energetics of nucleation or self-assembly step as given in (iii). The nucleation or self-assembly depends on the nature of building blocks. These building blocks contain a limited and countable number of atoms. Various combinations of interactions, often with feedback such as chemical bonding, electrostatic attraction or repulsion, mechanical stress, and spatial confinement come into play to generate a specific reaction field at a defined length scale. In this way, macroscopic hierarchical structures can be spontaneously built up from assemblies of smaller units formed at the nanoscale. Now, the formation of nanocrystals having a particular shape depends on the interaction between the building blocks in the self-assembling step. So, by controlling the interaction between the building blocks, one can control the shape of the nanocrystal. This control of the interaction between the building blocks can be achieved in two ways. The first way consists of using surfactants, a mixture of surfactants, or tuning of the temperature. Thus the interactions between the building blocks are different from one environment to the other. This results in different shapes of the nanocrystal. In the other method, one can control the interaction between the building blocks by preparing different kinds of building © 2007 American Chemical Society
16072 J. Phys. Chem. C, Vol. 111, No. 44, 2007
Letters
TABLE 1: Schematic Representation of Four Different Cases case
diffusion couple
difference of diffusion rate
case I case II case III case IV
(S,X) (S,Y) (S,A) (S,B)
M M1 M2 M3
TABLE 2: Structural Units for Four Different Cases case
structural units/building blocks
case I case II case III case IV
ZnS Zn2S2 Zn5S5 Zn8S8
blocks, that is, building blocks of different sizes and/or shapes. So, if different kinds of building blocks can be prepared by some mechanism, they will interact in a different way in the self-assembling step, and, as a result, one will get nanocrystals of different shapes. However, the nature of the building blocks depends on the rate of diffusion. For example, in the formation of ZnS nanocrystals, the diffusion rate of S into Zn solution plays a crucial role. As we are interested in the role of the diffusion couple in determining the shape of a nanocrystals, we assume that with sulfur there must be present another component that also diffuses in the Zn solution. The role of a chosen diffusion couple can be understood by working with different couples whose diffusion rates in the suspension of Zn nanocrystals are different and calculating the activation energy for the formation of ZnS nanocrystals of a particular shape. Let us consider the cases of four different couples, say, (S,X), (S,Y), (S,A), and (S,B). In the first case, there are two components, one is sulfur and other is X (say), and both are diffusing into Zn solution. The diffusion rates of sulfur and X through the suspension of Zn crystals are of considerable amount. In the second case, there are two components, one is again sulfur and other is Y (say). The difference of diffusion rates in the Zn solution between sulfur and Y are larger than the difference of diffusion rates in Zn solution between sulfur and X. In the third case, there are two components, one is sulfur and other is A; both are diffusing in to suspension of Zn crystals, and the difference of diffusion rates between sulfur and A is large compared to first and second cases. In the fourth case, the two components are sulfur and B, and the difference of diffusion rates in Zn crystals between sulfur and B is very large. So we assume that the difference of diffusion rates between their components increases as we go from (S,X) to (S,B). We show this schematically in Table 1. In Table 1 M is used to indicate that the difference of diffusion rate between the components of the couple is only a considerable amount, and the difference of diffusion rates are M3 > M2 > M1 > M. As we are interested in the formation of ZnS nanocrystals, we have assumed that the diffusion rate of S in Zn crystals is higher than that of the other components in a diffusion couple. Let us try to qualitatively understand the implication of having a difference of diffusion rates between the components of a diffusion couple in the formation of building blocks or structural units. In the first case, where both S and X are diffusing in a suspension of Zn nanocrystals, several ZnS monomer units are prepared along with ZnX units. The role of the diffusion partner X here is to prevent the formation of larger ZnS units since X is competing with S for binding with Zn. The ZnS units that are formed can be considered as building blocks. Let us consider the second case where we assumed that the difference in
Figure 1. (A) Energy profile of the formation of bubble and zincblende clusters for case I of Table 1. (B) Initial, transition, and final structures corresponding to panel A.
diffusion rates is M1 > M. As a result, the number of S atoms diffusing into the Zn crystals per unit time will be greater than that in the first case. Understandably, in this case, one will expect to have a larger structural unit, say, Zn2S2 unit at a time. This means that, after the diffusion step, there are several Zn2S2 units are present in the system, and each Zn2S2 unit is assembled one by one to form the large crystal. Since M2 > M1, the structural units formed in the third case will be larger than those formed in case II, say, Zn5S5. Similarly, the structural units in
Letters the fourth case will be the largest, say, Zn8S8. The reason behind the choice of this structural unit is that there is experimental evidence that molecules and small clusters can act as elementary growth units for the formation of large clusters.18,19 The different structural units/ building blocks for the four different cases are given in Table 2. We have chosen the initial structure of these building blocks and their arrangements in such a way that they remain in minimum force. This choice of the initial structure of the building blocks also requires the least activation energy for the formation of large clusters. Let us suppose that we would like to prepare a Zn40S40 zincblende or hollow structure. So when nucleation takes place, these structural units (e.g., in case I, ZnS, in case II, Zn2S2, in case III, Zn5S5, and in case IV, Zn8S8), which are a considerable distance from each other, are assembled one by one at some nucleation center to form the large crystal. Clearly, in case I, there are 40 structural units to form the Zn40S40 cluster. In the second, third, and fourth cases, 20, 8, and 5 structural units will form the large (Zn40S40) cluster. The final cluster may either be a zinc-blende or a hollow one. One may be interested to see which of these final shapes is much more probable when the difference of diffusion rate changes from case I to case IV. To that end, we have calculated the minimum energy path (MEP) for the formation of either a zinc-blende or hollow structure by using the nudged elastic band (NEB) method.20-22 The NEB method provides an efficient recipe for finding the MEP between a given initial and final state of transition. The MEP is found by constructing a set of images of the system between the initial and final states. A spring interaction between adjacent images is added to ensure continuity of the path, thus making an elastic band. An optimization of the band involving the minimization of the force acting on the images brings the band to the MEP. We used the density functional tight-binding (DFTB)23,24 method for energy minimization. Because the DFTB method is parametrized, we tested its accuracy by calculating the structural parameters for an infinite periodic crystalline form, and the structural parameters are in good agreement with the experimental values.17 For case I, we have shown the minimum path so obtained in Figure 1A. Figure 1B displays the initial, transition, and final structures for this case. From Figure 1A, it is clear that the activation barrier for the nucleation of ZnS structural units to form a hollow structure is less than that of the zinc-blende structure by an amount of 0.123 eV. The other interesting feature is that the formation of both a hollow or zincblende structure is accompanied by the release of energy. The idea that the formation is an exothermic process is in agreement with the experiment.25 The activation barriers for the nucleation process in the second case of the formation of both zinc-blende and hollow structures are shown in Figure 2A. As before the initial, transition, and final structures in this case are shown in Figure 2B. The activation barrier for the formation of hollow crystal is less than that of zinc-blende crystal by an amount of 0.04 eV. The formation of hollow or zinc-blende crystal is again an exothermic process. For case III, the activation barrier for the nucleation step is shown in Figure 3A, and the corresponding initial, transition, and final structures are shown in Figure 3B. From the figure it is seen that the activation barrier for zincblende crystal is less than that for hollow crystal by an amount of 0.4475 eV. So when the difference of diffusion rates between the diffusion couple is large, the formation of zinc-blende crystal is more favorable than the formation of hollow crystal. The MEP for the nucleation steps in the last case are shown in Figure 4A, and the corresponding initial, transition, and final structures are shown in Figure 4B. From this figure it is evident that, when
J. Phys. Chem. C, Vol. 111, No. 44, 2007 16073
Figure 2. (A) Energy profile of the formation of bubble and zincblende clusters for case II of Table 1. (B) Initial, transition, and final structures corresponding to panel A.
the difference of diffusion rate between the diffusion couple is very large, the formation of zinc-blende crystal is energetically more favorable than the formation of hollow crystal. The nucleation barrier height for the zinc-blende crystal form is less than that of the hollow crystal form by an amount of 0.544 eV. Another interesting feature that emerges from all these figures
16074 J. Phys. Chem. C, Vol. 111, No. 44, 2007
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Figure 3. (A) Energy profile of the formation of bubble and zincblende clusters for case III of Table 1. (B) Initial, transition, and final structures corresponding to panel A.
is that the magnitudes of ∆H decrease from case I to case IV, and this has one important implication. In all four cases we have described, the formation of either hollow or zinc-blende clusters was accompanied by a decrease in entropy, that is, ∆S is negative. The value of ∆S is very negative for the first case, and the magnitude decreases as we go from first to fourth. The formation of nanocrystals is spontaneous, so ∆G is always negative. Now because ∆S is a large negative for case I, ∆H should be largest in magnitude to maintain a negative ∆G, and ∆H decreases from case I to case IV.
Figure 4. (A) Energy profile of the formation of bubble and zincblende clusters for case IV of Table 1. (B) Initial, transition, and final structures corresponding to panel A.
In conclusion, we have studied the mechanism and energetics of the formation of hollow and zinc-blende nanocrystals. Although we have considered ZnS as a prototypical example, our results are general and valid for other nanocrystals. The formation of nanocrystals of a particular shape, either zincblende or hollow, depends on the nature of the structural units or building blocks, which is determined by the diffusion rates
Letters of the diffusion couple. We believe that our theoretical results will help provide better theoretical understanding of the experimental observation of Yin et al.11 For the formation of hollow nanocrystals, the building blocks should contain fewer ZnS units, which results only when the difference of diffusion rates in a diffusion couple is of a considerable amount but not too high. This is also physically realizable since, for the formation of a hollow crystal, all the structural units or building blocks have to be assembled on the surface. This is possible only when the structural units contain a fewer number of atoms. If the difference of diffusion rates in the diffusion couple is much larger, then the structural unit or building blocks will contain larger ZnS units, and the formation of a bulk-like structure will be energetically more favorable. Thus, by proper choice of the diffusion partner, one can control the building blocks or structural units and hence the shape of a particular nanocrystal. The choice of different initial structures can only affect the magnitude of the activation energy, but our final conclusions will remain the same. Acknowledgment. The financial support from CSIR, Govt. of India, and UGC (SAP), New Delhi, through research grants is gratefully acknowledged. The authors would like to thank Prof. B. Talukdar, Dept. of Physics, Visva-Bharati, for many useful discussions. References and Notes (1) Rao, C. N. R.; Kulkarni, G. U.; Thomas, P. J.; Edwards, P. P. Chem.sEur. J. 2002, 8, 28.
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