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Geometric thermal phase diagrams for studying the thermal dynamic stability of hollow gold nanoballs at different temperatures Luyun Jiang, Wei Sun, Yajun Gao and Jianwei Zhao* Thermal stability is one of the main concerns for the synthesis of hollow nanoparticles. In this work, molecular dynamics simulation gave an insight into the atomic reconstruction and energy evolution during the collapse of hollow gold nanoballs, based on which a mechanism was proposed. The stability
Received 24th November 2013, Accepted 14th January 2014
was found to depend on temperature, its wall thickness and aspect ratio to a great extent. The relationship among these three factors was revealed in geometric thermal phase diagrams (GTPDs). The
DOI: 10.1039/c3cp54961e
GTPDs were studied theoretically, and the boundary between different stability regions can be fitted and calculated. Therefore, the GTPDs at different temperatures can be deduced and used as a guide for
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hollow structure synthesis.
1 Introduction Nanomaterials underwent a booming development in the past decade and became a new separate branch of chemistry. Nowadays, it is possible for scientists to control the size, shape and component of nanoparticles in order to achieve various properties and enable different implementations. Compared with various solid nanoparticles, hollow structures exhibit unique properties including an inner nanocage, relatively low densities, high surface-to-bulk ratio, and special photonics properties, which lead to wide applications.1–5 With the hollow structure, hollow gold nanoballs (HGNBs) can serve as extremely small containers in applications related to catalysis, drug delivery, cancer hyperthermia and protection of environmentally sensitive materials.6–10 Owing to their special photonics properties, like surface-enhanced Raman scattering (SERS), HGNBs are also promising materials in photo-thermal therapy, optoelectronic devices and calorimetric sensing.6,11–13 These intrinsic properties and applications are highly dependent on the shape, size and shell thickness of the HGNB.2–3,12,14 It is found that the surface plasmon resonance (SPR) of HGNBs can be smoothly tuned across the visible and near-infrared spectral region by varying the aspect ratio, which is sometimes defined as the outer diameter/shell thickness.7,15–17 Recently, Chandra, and Knappenberger, Jr. et al. reported that HGNBs displayed larger first hyperpolarizabilities than solid gold nanospheres, which enables their usage in various nonlinear microscopies.11 Key Laboratory of Analytical Chemistry for Life Science, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210008, P. R. China. E-mail: [email protected]; Tel: +86-25-83596523
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And a size-dependent coherent acoustic vibration in HGNBs has also been reported.18 HGNBs are usually synthesized by two methods. One is coating the desired materials onto a template surface, followed by calcinations or wet etching to remove the cores. Another approach is by one-step galvanic replacement reactions from sacrificial templates,6–7,15 which was first developed by Sun and co-workers.19 According to this method, well-defined void sizes and homogeneous, smooth, high crystalline walls could be generated.7 One of the main concerns involved in the synthesis process is the temperature control, as highlighted in the various literatures.7,15,20–22 Temperature is of great importance in the synthesis process because it highly affects the thermal stability of the hollow structure and therefore limits the size and shape.17,23 It is widely considered that temperature plays the most important role in determining the final morphology of HGNBs.7,15 Yin et al. pointed out that an Au shell surrounding a 4 nm void should remain hollow for decades if the temperature is held below 150 1C.24 Knappenberger et al. stated that important future experiments include a systematic study of the dependence of inter-particles coupling on the particles’ aspect ratio.16 Therefore, the direct dependence of structure on temperature is well worth quantitative investigation. Computer simulation has proved to be a powerful tool for this purpose. Chookajorn et al.25 developed a nanostructure stability map based on a thermodynamic model to design stable nanostructured tungsten alloys and has shown successes in experiments. In our previous work,26,27 the stability of HGNBs with different aspect ratios, defined as outer diameter/shell thickness, and wall thicknesses at 300 K were studied and summarized in a two-dimension figure,
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called geometric thermal phase diagrams (GTPDs). The basic idea behind this method is that a HGNB is initially simulated with given aspect ratio (taken as the y axis in the GTPD) and wall-thickness (taken as the x axis in the GTPD), then relaxed. The final structure of the HGNB decides which category of the stability it should belong to. The categories including stable (unchanged structure), semi-stable (collapse, but maintain a hollow structure) and unstable (collapse into a solid structure). In this work, we develop the GTPD at 300 K to a wider temperature range from 4 to 600 K. The vast systems simulated comprise GTPDs under temperatures of 4, 150, 300, 450 and 600 K, and shows the areas in GTPDs as a function of temperature, which enable the deduction of the GTPDs at any specific temperature. The GTPDs can find applications in the optimisation between temperature and the HGNB structure which can be wellcontrolled by the size of the templates and the amount of gold salt.12,15 In addition, the limitations of experimental synthesis and application are shown. For example, in order to delivery drugs in the human body, the inner space of the HGNB is largely required to be the medicine’s size. By using the GTPD at body temperature, the required minimum outer diameter and wall-thickness can be calculated.
Molecular Dynamics (MD) is a simulation method originally developed from theoretical physics in the late 1950s, which can give a view of molecular motion on an atomic scale. For computer simulations, the vast numbers of particles in molecular systems always cause problems for analysing the required properties. MD circumvents this by numerical approximation with the proper selection of ensemble, force field, algorithms and parameters.28,29 This work was carried out under a Canonical ensemble system (NVT) where the temperature of the system is kept constant by using the Nos´e–Hoover thermostat as a velocity rescaling method.30–32 There are two common used force fields in MD simulations, which are pair potential and many-body potential. Instead of calculating the interaction between all the particles of the system, we used the embedded-atom method (EAM) to describe the interaction, a many-body potential based on Density Functional Theory (DFT).33,34 In EAM theory developed by Johnson,35,36 the electron density in the region of an atom is only contributed to from surrounding atoms, and then as a function of the electron density, the potential energy can be calculated.37,38 EAM has been proved to be successful for pure metallic systems with no directional bonding, for example, Au, Cu and Ag.37,39 For the metal of face-centered cubic (fcc) structures, the total energy was given by: 1X X V rij þ F ðri Þ 2 ij i
(1)
X j rij
(2)
ri ¼
3 Results and discussion 3.1
2 Methodology
E¼
the electronic density function and F(ri) is the embedding considering the effects from free electrons in the metal. All the MD simulations and visualizations were performed in a self-developed code, NanoMD, which has been used to study nanostretching, nanobreaking, nanostomic chain of various fcc structure metals including Ag, Au, and Cu.40,41 To optimise the calculation efficiency, NanoMD integrated parallel algorithms; the Verlet leapfrog algorithm and celllinked list algorithm were used. The centrosymmetry parameter (CSP) was also utilized to identify the local atomic structure. The free boundary condition was adopted. The code has been well demonstrated in former papers.42–44 Considering the simulation size, the time length to evaluate the potential should be carefully chosen. The integration time step in this work is 1.6 femtosecond, smaller than the fastest vibration frequency in the system to avoid discretization errors. Many experiments from the literatures relating to nanoparticle synthesis have been investigated and most of them were in accordance with our results. This means the choice of parameters in this work is reasonable.
Dynamic process of HGNBs at different temperatures
The geometric model of a HGNB was built in NanoMD as follows. First, an initial configuration was created with fcc structure lattices along the [100] crystallographic orientation, then the cubic was caved into a hollow structure with given values of the outer-diameter (R) and inner-diameter (r) as shown in Fig. 1. The momentum and coordinate space of each gold atom were recorded, from which the atom–atom interaction and potential energy of the HGNB were calculated. Then the HGNB system was relaxed until the potential energy reached equilibrium. Fig. 2 shows the potential energy of HGNBs with a 3 nm wall thickness and 14 nm diameter fluctuated at temperatures 4, 150, 300, 450 and 600 K respectively. The perfect single crystal structure gives a high potential energy at the beginning of all simulations (MD step = 0), and the initial energy is proportional to temperature. Then it drops sharply when relaxed (MD steps from 0 to 5 104) and keeps decreasing gradually (MD steps from 5 104 to 3 105), corresponding to the HGNB structure adjustment. Finally, the potential energy reaches equilibrium (MD steps after 3 105), which suggests a stable state for
ij
where E is the total internal energy, V(rij) is the pair interaction energy between an atom i and its neighboring atom j, j(rij) is
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Fig. 1
Visual images of initial state of hollow gold nanoball.
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Fig. 2 The evolution of potential energy of HGNBs with 3 nm wall thickness and 14 nm diameter, during 400 000 MD simulation steps at temperatures of 4, 150, 300, 450 and 600 K.
the HGNB. At 4 K, the potential energy of the HGNB fluctuates around the initial value during the whole simulation process, which indicates the original structure was kept perfectly. In this case the atomic rearrangement can hardly happen because at low temperatures the thermal energy of the gold particles is insufficient to overcome the energy barrier of slipping. In 150, 300, 450 and 600 K, the increasing thermal energy allows gold atoms to adjust their structure to achieve a lower potential energy state. It can be estimated that the energy of the system to should be between 0.475 V (4 K) to 0.46 V (150 K). At 600 K the potential energy reaches a plateau before achieving the equilibrium. This indicates that the system underwent a metastable state, which can be attributed to the thermal energy being so high that the gold particles move quickly enough to oppose the restructuring. The meta-stable state also relates to the size of the HGNBs. Fig. 3 shows potential energy evolution of HGNBs at 600 K with thickness of 3.5 nm and varying diameters of 24.5, 28 and 31.5 nm. It can be observed that a meta-stable plateau starts from approximate 9 104 MD steps for all the systems, suggesting that it is only a function of temperature but not diameter. However, the lengths of plateau are different. The meta-stable state ends at about 2.6 105 MD step for smallest HGNB system, which is four times longer than the HGNB of 28 nm diameter (plateau ends at 1.3 105 MD steps). For the HGNB of 31.5 nm, the semi-stable
Fig. 3 The evolution of potential energy of HGNBs with 3.5 nm wall thickness and different diameter values of 24.5, 28 and 31.5 nm at 600 K.
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plateau is hardly observed. This simulation result suggests that with the same wall thickness, the HGNBs of smaller diameter might undergo a longer meta-stable state. Corresponding to the potential energy curves in Fig. 2, Fig. 4 shows snapshots during the simulation process for a HGNB of 3 nm wall thickness and 14 nm diameter at 4, 150, 300, 450 and 600 K respectively. Those snapshots were captured as projections of gold atoms in the z plane. The first row snapshots are the initial structures at various temperatures. It can be observed that the atom arrangement gets less regular as the temperature increases because the atom vibration is fiercer. The structure re-arrangement at each temperature can be seen in the columns. At 4 K, the last snapshot, showing the projection after equilibrium, is similar to the initial one. It suggests that at 4 K the HGNB kept its initial structure during the whole relaxing process. At 150 K, as shown in the second column, the atoms of HGNB started to move and be reconstructed at [111] then the disorder spread to [110] and [100] gradually. Finally, the HGNB collapses to an irregular hollow structure, as shown in the last snapshot in column 2. More details regardinf this collapse process were given in the previous paper, and the cross-sectional images and animation gif files were attached in its supporting information.26 The result that the collapse started from [111] facet matches other experiments. Finbow et al. reported that at low temperatures nanoscale metal wires were found to slip over a [111] plane.45 The reason might be that for a close-packed structure such as fcc, the smallest Burgers vectors exist along the [110] direction, which makes it energetically favorable to reconstruct along the (111) slip planes. The similar collapse processes could be observed at 300 and 450 K in the third and forth column respectively. In 600 K, the HGNB collapsed isotropically, and maintained a semistructure in the process which corresponded to the meta-stable state in Fig. 2. And at the end, the HGNB it 600 K collapsed into a solid structure. In order to classify the HGNB systems, we defined three categories of sample stability, i.e. stable, meta-stable and unstable. A stable HGNB is one which can keep the initial potential energy
Fig. 4 Representative snapshots during the MD simulation process for the HGNBs with 3 nm wall thickness and 14 nm diameter at 4, 150, 300, 450 and 600 K.
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and atom structure, as does the HGNB of 3 nm wall thickness and 14 nm diameter at 4 K as shown in Fig. 2 and 4. A semistable HGNB is one in which the initial structure would collapse into an irregular hollow structure, such as HGNBs at 150, 300 and 450 K. For unstable HGNBs, these would collapse into a solid structure, as does the one at 600 K.
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3.2 Geometric thermal phase diagrams at different temperatures In our previous work,26,27 the aspect ratio is defined by p = R/d
(3)
where p is the aspect ratio, R is the outer radius, d is the wall thickness. Therefore, a HGNB would be described by d and p. As mentioned above, the geometric thermal phase diagram (GTPD) was developed by taking the aspect ratio (a.r.) as the y axis and wall-thickness (w.t.) as the x axis, which can be seen in Fig. 5. Each point in the graph stands for a HGNB system with a certain wall-thickness and aspect ratio. According to the three categories of stability discussed before, those points are shown as a square, circle or triangle representing stable, semi-stable and unstable HGNB systems respectively. In the previous work,26 at GTPD at 300 K was drawn, as shown in Fig. 5c. It has been well demonstrated that the thermal stability of HGNB depends on the a.r. and w.t. to a great extent. With the same aspect ratio, the HGNB with a larger w.t. is more stable. But with the same w.t., an increasing a.r. leads to a less stable HGNB. In this work, we are focused on how the temperature influences the stability of HGNB. The GTPDs at 4, 150, 450 and 600 K are shown in Fig. 5a, b, d and e respectively. In the 4 K GTPD (Fig. 5a), there is no unstable systems in the region studied, and the region of stable systems starts from a w.t. of 1.25 nm. The reason for the high stability is that the temperature is too low to provide enough kinetic energy for gold atoms to cross the energy barriers of atom slip and dislocation. At 150 K (Fig. 5b), the stable region starts from 2.0 nm w.t. and unstable systems can be found at 0.5–1.0 nm w.t. and 6–2 a.r. At 300 K (Fig. 5c), the stable system can only be found for w.t. larger than 2.5 nm and the unstable region enlarges to 1.5 nm. For 450 and 600 K (Fig. 5d and e), there is no stable system in the range studied, and the area of unstable HGNBs increases. To sum up, with increasing temperature, there are less stable systems and more unstable systems in the GTPDs. The areas of stable/semi-stable/unstable HGNB all move towards larger wall-thickness as the temperature rises. In other words, the stability of the HGNBs decreases. The decrease in stability is more obvious if we trace a certain size of HGNBs at increasing temperatures. For example, one HGNB with 8 a.r. and 1.5 nm w.t., was a stable system at 4 K which kept its initial structure. When the temperature increased to 150, 300 and 450 K, the HGNB underwent a structural re-arrangement and collapsed into an irregular hollow structure, appearing as a semi-stable system in the GTPD. At 600 K, the HGNB collapsed into a solid which indicated an unstable system.
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Fig. 5 The geometric thermal phase diagrams at temperatures of (a) 4 K, (b) 150 K, (c) 300 K, (d) 450 K and (e) 600 K. The boundary isometric line in b, c, d and e were fitted by eqn (4). The square, circle and triangle stand for stable, semi-stable and unstable HGNBs respectively.
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3.3 Boundaries and prediction of geometric thermal phase diagrams Limited by the computational capability, the simulations were carried out for the w.t. ranged from 0.5–3.0 nm and the a.r. from 2–18. However, by analysing the different areas and boundaries mathematically, we are able to predict the GTPDs beyond the computed range. An unstable HGNB is the one which collapses from its initial hollow structure gold sphere into an irregular solid ball. Imagine an ideal and ‘reversible’ collapse process wherein at each step the regular HGNB re-structures to another regular HGNB, until it reaches a regular solid sphere. In this situation, each rearranged HGNB can be presented as a point in the GTPD. All the points will compose a line segment. This line segment can be considered as part of a line, called an isometric line. All the points in this line share the same number of gold atoms. Since the gold atom number of a HGNB is kept constant, the isometric line can be described by eqn (4), as deduced in our previous work.26 d3[3p2 3p + 1] = C(T)
(4)
where C is a temperature-dependent constant (see below), p is the aspect ratio, and d is the wall thickness. In an ideal system, every unstable HGNB will collapse following an isometric line into a solid. In other words, if one point in the isometric line is found to be an unstable system, all the points in this line should also be unstable systems, sharing the same atom re-structure pathway. The unstable area is therefore composed of those lines. In a GTPD, HGNBs between unstable and semi-stable regions are close to the equilibrium, therefore these approach the ideal system if the step time is chosen small enough. In this case, the boundary line can be roughly fitted by eqn (4). Take the GTPD at 150 K for example, a point of 3 a.r. and 0.75 w.t. was chosen, which is between unstable points and semi-stable points, then the constant C can be calculated to be 8. An isometric line is therefore can be drawn in the GTPD, as displayed in Fig. 5b. By varying the point value in the area between unstable region and semi-stable region, different C values are obtained, as well as isometric lines. The one which fits the boundary most accurately should be chosen. In this way, the boundary line is achieved. The fitting lines of GTPDs in 150, 300, 450 and 600 K are shown in Fig. 5b–e, of which the C are found to be 8, 64, 120 and 1000 respectively. All the boundary fitting lines are drawn together in Fig. 6. It is apparent that with increasing temperature, the boundary fitting line moves toward higher a.r. and thicker w.t. Another thing that needs to be pointed out is that at lower temperatures, the accuracy and precision of the fitting of the boundary in the GTPD is better. In high temperatures, i.e. 600 K in Fig. 5e, it is hard to find an isometric line to fit to the boundary between the unstable region and semistable region. It is probably because the higher temperature drives the systems far away from the ideal equilibrium condition; therefore this method cannot be used. So far there is no proper theory that can be used to fit the boundary between
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Fig. 6 The fitting boundary isometric line in geometric thermal phase diagrams at 150, 300, 450 and 600 K with C equal to 8, 64, 120 and 1000 respectively.
semi-stable systems and stable systems. On the other hand, the boundaries between semi-stable systems and unstable systems, which are the focus of this work, are of more importance in practice. As reported experimentally, most of the hollow gold nanoballs (HGNBs) are semi-stable or unstable systems and the difference between them (having a hollow structure inside) is the major character of HGNBs and limits most of the applications such as drug delivery. As mentioned above, the value of constant C is a function of temperature. To investigate the relationship between them, more GTPDs in the temperature range of 150 to 700 K were simulated, drawn and fitted, where more values of the constant C were obtained. The reason to choose this temperature range is that there is hardly any unstable systems in the GTPDs at lower temperatures than 150 K (Fig. 5b), and it is difficult to accurately and precisely fit the GTPDs and find relationship between C and temperature at higher temperatures, as discussed previously. The plot of the logarithm of C against the inverse of temperature is displayed in Fig. 7, yielding two linear lines with different slopes. One is in the range from 150 to 450 K, corresponding to 0.0022 to 0.0067 in Fig. 7, and the other one is from 450 to 700 K, corresponding to 0.0022 to 0.0014. Therefore, in this range the constant C can be calculated if the temperature is known, and the boundary between unstable and
Fig. 7 The relationship between temperature and the constant in isometric line equation, 1/T vs. ln C.
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semi-stable areas in the GTPD can be deduced. Consequently the GTPD at this temperature can be predicted. The turning point of 450 K is of great interest. Although the physical meaning of this point is hard to fully understand, some explanation can be given. As seen from Fig. 2, the system energy at 450 K is close to the energy needed to slip. Therefore, the systems at higher temperatures may undergo a different structural re-arrangement from those at lower temperatures. In particular, at low temperature, the slipping along the (111) facet in the crystal hollow ball is the main mechanism of structural collapse. On the other hand, an isotropic structural collapse becomes the dominant mechanism at high temperatures. This might be the origin of the switch between two lines in Fig. 7. The GTPDs are in agreement to the experimental results from the literature.7,15–17 To the best of our knowledge, the HGNBs reported are all in the region of stable/semi-stable systems at 300 K (Fig. 5c). Dowgiallo et al. reported HGNBs with aspect ratios from 1.6 to 4 and wall thicknesses from 3 to 11 nm.18 This is in the stable region of the GTPD at 300 K (Fig. 5c), showing a high stability at room temperature. However, this HGNB should not be heated to over 450 K in experiment since the HGNB may be in the semi-stable region (Fig. 5d). And if the temperature is higher than 600 K, the HGNB may collapse into a solid sphere (Fig. 5e).
4 Conclusion In this work, we investigated how the temperature and the geometric features (wall thickness and aspect ratio) would affect behaviours of HGNBs. The increasing temperature leads to higher average atomic kinetic energy, then less stability and at last faster collapse processes. However, when the temperature is high enough, a semi-stable stage appears, corresponding to a homogeneous collapse process. The geometric thermal phase diagrams, which indicates the thermal dynamic stability of a HGNB with a certain wall-thickness and aspect ratio, was developed over a wider temperature range from 4 to 600 K. As temperature increased, the unstable region in the GTPD is enlarged. The boundaries between the unstable and semi-stable regions in GTPDs over the temperature range from 4 to 600 K were fitted. The relationship between the boundary line and temperature enables us to propose probable GTPDs at other temperatures, and predict the thermal stability of HGNBs at different temperatures.
Acknowledgements This work was funded by National Key Technology R&D Program of China 2012BAF03B05 and the National Natural Science Foundation of China (NSFC) No. 21121091, 21273113, and 51071084.
References 1 H. C. Zeng, J. Mater. Chem., 2006, 16, 649. 2 J. Meier, J. Schiotz, P. Liu, J. K. Norskov and U. Stimming, Chem. Phys. Lett., 2004, 390, 440.
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3 S. Link and M. A. El-Sayed, Int. Rev. Phys. Chem., 2000, 19, 409. 4 Y. Ding, F. R. Fan, Z. Q. Tian and Z. L. Wang, J. Am. Chem. Soc., 2010, 132, 12480. 5 V. Uzayisenga, X. D. Lin, L. M. Li, J. R. Anema, Z. L. Yang, Y. F. Huang, H. X. Lin, S. B. Li, J. F. Li and Z. Q. Tian, Langmuir, 2012, 28, 9140. 6 Y. G. Sun, B. Gates, B. Mayers and Y. N. Xia, Nano Lett., 2002, 2, 165. 7 Y. G. Sun, B. Mayers and Y. N. Xia, Adv. Mater., 2003, 15, 641. 8 P. R. Selvakannan and M. Sastry, Chem. Commun., 2005, 1684. 9 X. Sun, C. Zhai and X. Y. Wang, Bioprocess Biosyst. Eng., 2013, 36, 273. 10 W. Zhu, T. Chen, X. Ma, H. Ma and S. Chen, Colloids Surf., B, 2013, 111, 321. 11 M. Chandra, A. M. Dowgiallo and K. L. Knappenberger, J. Phys. Chem. C, 2010, 114, 19971. 12 A. M. Schwartzberg, T. Y. Olson, C. E. Talley and J. Z. Zhang, J. Phys. Chem. B, 2006, 110, 19935. 13 J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, D. Y. Wu, B. Ren, Z. L. Wang and Z. Q. Tian, Nature, 2010, 464, 392. 14 M. Chandra, A. M. Dowgiallo and K. L. Knappenberger, J. Am. Chem. Soc., 2010, 132, 15782. 15 Y. G. Sun and Y. N. Xia, J. Am. Chem. Soc., 2004, 126, 3892. 16 K. L. Knappenberger, A. M. Schwartzberg, A. M. Dowgiallo and C. A. Lowman, J. Am. Chem. Soc., 2009, 131, 13892. 17 S. Preciado-Flores, D. C. Wang, D. A. Wheeler, R. Newhouse, J. K. Hensel, A. Schwartzberg, L. H. Wang, J. J. Zhu, M. Barboza-Flores and J. Z. Zhang, J. Mater. Chem., 2011, 21, 2344. 18 A. M. Dowgiallo, A. M. Schwartzberg and K. L. Knappenberger, Nano Lett., 2011, 11, 3258. 19 C. W. Huang, J. C. Jiang, M. Y. Lu, L. Sun, E. I. Meletis and Y. W. Hao, Nano Lett., 2009, 9, 4297. 20 S. N. Abdollahi, M. Naderi and G. Amoabediny, Colloids Surf., A, 2013, 436, 1069. 21 J. H. Lin, W. D. He, S. Vilayurganapathy, S. J. Peppernick, B. Wang, S. Palepu, M. Remec, W. P. Hess, A. B. Hmelo, S. T. Pantelides and J. H. Dickerson, ACS Appl. Mater. Interfaces, 2013, 5, 11590. 22 B. S. Gutrath, M. F. Beckmann, A. Buchkremer, T. Eckert, J. Timper, A. Leifert, W. Richtering, G. Schmitz and U. Simon, Nanotechnology, 2012, 23, 225707. 23 G. Schneider and G. Decher, Nano Lett., 2004, 4, 1833. 24 Y. D. Yin, C. Erdonmez, S. Aloni and A. P. Alivisatos, J. Am. Chem. Soc., 2006, 128, 12671. 25 T. Chookajorn, H. A. Murdoch and C. A. Schuh, Science, 2012, 337, 951. 26 L. Y. Jiang, X. Yin, J. W. Zhao, H. M. Liu, Y. H. Liu, F. Y. Wang, J. J. Zhu, F. Boey and H. Zhang, J. Phys. Chem. C, 2009, 113, 20193. 27 L. Y. Jiang, X. Yin and J. W. Zhao, Chin. J. Inorg. Chem., 2009, 25, 176. 28 A. Rahman, Phys. Rev., 1964, 136, A405.
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29 30 31 32
Published on 16 January 2014. Downloaded by Arizona State University on 30/10/2014 18:13:43.
33 34 35 36 37 38
B. J. Alder and T. E. Wainwright, J. Chem. Phys., 1959, 31, 459. S. Nose, Mol. Phys., 1984, 52, 255. S. Nose, J. Chem. Phys., 1984, 81, 511. F. Y. Wang, Y. H. Liu, T. M. Zhu, Y. J. Gao and J. W. Zhao, Nanoscale, 2010, 2, 2818. A. D. Becke, J. Chem. Phys., 1993, 98, 5648. A. D. Becke, Phys. Rev. A: At., Mol., Opt. Phys., 1988, 38, 3098. R. A. Johnson, Phys. Rev. B: Condens. Matter Mater. Phys., 1988, 37, 3924. R. A. Johnson, Phys. Rev. B: Condens. Matter Mater. Phys., 1989, 39, 12554. M. S. Daw, S. M. Foiles and M. I. Baskes, Mater. Sci. Rep., 1993, 9, 251. M. S. Daw and M. I. Baskes, Phys. Rev. B: Condens. Matter Mater. Phys., 1984, 29, 6443.
This journal is © the Owner Societies 2014
Paper
39 M. S. Daw and M. I. Baskes, Phys. Rev. B: Condens. Matter Mater. Phys., 1984, 29, 6443. 40 F. Y. Wang, T. N. Chen, T. M. Zhu, Y. J. Gao and J. W. Zhao, J. Appl. Phys., 2011, 110, 1082. 41 Y. J. Gao, Y. Q. Fu, W. Sun, Y. L. Sun, H. B. Wang, F. Y. Wang and J. W. Zhao, Comput. Mater. Sci., 2012, 55, 322. 42 D. X. Wang, J. W. Zhao, S. Hu, X. Yin, S. Liang, Y. H. Liu and S. Y. Deng, Nano Lett., 2007, 1208. 43 Y. H. Liu, J. W. Zhao and F. Y. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 115417. 44 J. W. Zhao, K. Murakoshi, X. Yin, M. Kiguchi, Y. Guo, N. Wang, S. Liang and H. Liu, J. Phys. Chem., 2008, 112, 20088. 45 G. M. Finbow, R. M. LyndenBell and I. R. McDonald, Mol. Phys., 1997, 92, 705.
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Geometric thermal phase diagrams for studying the thermal dynamic stability of hollow gold nanoballs at different temperatures Luyun Jiang, Wei Sun, Yajun Gao and Jianwei Zhao* Thermal stability is one of the main concerns for the synthesis of hollow nanoparticles. In this work, molecular dynamics simulation gave an insight into the atomic reconstruction and energy evolution during the collapse of hollow gold nanoballs, based on which a mechanism was proposed. The stability
Received 24th November 2013, Accepted 14th January 2014
was found to depend on temperature, its wall thickness and aspect ratio to a great extent. The relationship among these three factors was revealed in geometric thermal phase diagrams (GTPDs). The
DOI: 10.1039/c3cp54961e
GTPDs were studied theoretically, and the boundary between different stability regions can be fitted and calculated. Therefore, the GTPDs at different temperatures can be deduced and used as a guide for
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hollow structure synthesis.
1 Introduction Nanomaterials underwent a booming development in the past decade and became a new separate branch of chemistry. Nowadays, it is possible for scientists to control the size, shape and component of nanoparticles in order to achieve various properties and enable different implementations. Compared with various solid nanoparticles, hollow structures exhibit unique properties including an inner nanocage, relatively low densities, high surface-to-bulk ratio, and special photonics properties, which lead to wide applications.1–5 With the hollow structure, hollow gold nanoballs (HGNBs) can serve as extremely small containers in applications related to catalysis, drug delivery, cancer hyperthermia and protection of environmentally sensitive materials.6–10 Owing to their special photonics properties, like surface-enhanced Raman scattering (SERS), HGNBs are also promising materials in photo-thermal therapy, optoelectronic devices and calorimetric sensing.6,11–13 These intrinsic properties and applications are highly dependent on the shape, size and shell thickness of the HGNB.2–3,12,14 It is found that the surface plasmon resonance (SPR) of HGNBs can be smoothly tuned across the visible and near-infrared spectral region by varying the aspect ratio, which is sometimes defined as the outer diameter/shell thickness.7,15–17 Recently, Chandra, and Knappenberger, Jr. et al. reported that HGNBs displayed larger first hyperpolarizabilities than solid gold nanospheres, which enables their usage in various nonlinear microscopies.11 Key Laboratory of Analytical Chemistry for Life Science, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210008, P. R. China. E-mail: [email protected]; Tel: +86-25-83596523
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And a size-dependent coherent acoustic vibration in HGNBs has also been reported.18 HGNBs are usually synthesized by two methods. One is coating the desired materials onto a template surface, followed by calcinations or wet etching to remove the cores. Another approach is by one-step galvanic replacement reactions from sacrificial templates,6–7,15 which was first developed by Sun and co-workers.19 According to this method, well-defined void sizes and homogeneous, smooth, high crystalline walls could be generated.7 One of the main concerns involved in the synthesis process is the temperature control, as highlighted in the various literatures.7,15,20–22 Temperature is of great importance in the synthesis process because it highly affects the thermal stability of the hollow structure and therefore limits the size and shape.17,23 It is widely considered that temperature plays the most important role in determining the final morphology of HGNBs.7,15 Yin et al. pointed out that an Au shell surrounding a 4 nm void should remain hollow for decades if the temperature is held below 150 1C.24 Knappenberger et al. stated that important future experiments include a systematic study of the dependence of inter-particles coupling on the particles’ aspect ratio.16 Therefore, the direct dependence of structure on temperature is well worth quantitative investigation. Computer simulation has proved to be a powerful tool for this purpose. Chookajorn et al.25 developed a nanostructure stability map based on a thermodynamic model to design stable nanostructured tungsten alloys and has shown successes in experiments. In our previous work,26,27 the stability of HGNBs with different aspect ratios, defined as outer diameter/shell thickness, and wall thicknesses at 300 K were studied and summarized in a two-dimension figure,
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called geometric thermal phase diagrams (GTPDs). The basic idea behind this method is that a HGNB is initially simulated with given aspect ratio (taken as the y axis in the GTPD) and wall-thickness (taken as the x axis in the GTPD), then relaxed. The final structure of the HGNB decides which category of the stability it should belong to. The categories including stable (unchanged structure), semi-stable (collapse, but maintain a hollow structure) and unstable (collapse into a solid structure). In this work, we develop the GTPD at 300 K to a wider temperature range from 4 to 600 K. The vast systems simulated comprise GTPDs under temperatures of 4, 150, 300, 450 and 600 K, and shows the areas in GTPDs as a function of temperature, which enable the deduction of the GTPDs at any specific temperature. The GTPDs can find applications in the optimisation between temperature and the HGNB structure which can be wellcontrolled by the size of the templates and the amount of gold salt.12,15 In addition, the limitations of experimental synthesis and application are shown. For example, in order to delivery drugs in the human body, the inner space of the HGNB is largely required to be the medicine’s size. By using the GTPD at body temperature, the required minimum outer diameter and wall-thickness can be calculated.
Molecular Dynamics (MD) is a simulation method originally developed from theoretical physics in the late 1950s, which can give a view of molecular motion on an atomic scale. For computer simulations, the vast numbers of particles in molecular systems always cause problems for analysing the required properties. MD circumvents this by numerical approximation with the proper selection of ensemble, force field, algorithms and parameters.28,29 This work was carried out under a Canonical ensemble system (NVT) where the temperature of the system is kept constant by using the Nos´e–Hoover thermostat as a velocity rescaling method.30–32 There are two common used force fields in MD simulations, which are pair potential and many-body potential. Instead of calculating the interaction between all the particles of the system, we used the embedded-atom method (EAM) to describe the interaction, a many-body potential based on Density Functional Theory (DFT).33,34 In EAM theory developed by Johnson,35,36 the electron density in the region of an atom is only contributed to from surrounding atoms, and then as a function of the electron density, the potential energy can be calculated.37,38 EAM has been proved to be successful for pure metallic systems with no directional bonding, for example, Au, Cu and Ag.37,39 For the metal of face-centered cubic (fcc) structures, the total energy was given by: 1X X V rij þ F ðri Þ 2 ij i
(1)
X j rij
(2)
ri ¼
3 Results and discussion 3.1
2 Methodology
E¼
the electronic density function and F(ri) is the embedding considering the effects from free electrons in the metal. All the MD simulations and visualizations were performed in a self-developed code, NanoMD, which has been used to study nanostretching, nanobreaking, nanostomic chain of various fcc structure metals including Ag, Au, and Cu.40,41 To optimise the calculation efficiency, NanoMD integrated parallel algorithms; the Verlet leapfrog algorithm and celllinked list algorithm were used. The centrosymmetry parameter (CSP) was also utilized to identify the local atomic structure. The free boundary condition was adopted. The code has been well demonstrated in former papers.42–44 Considering the simulation size, the time length to evaluate the potential should be carefully chosen. The integration time step in this work is 1.6 femtosecond, smaller than the fastest vibration frequency in the system to avoid discretization errors. Many experiments from the literatures relating to nanoparticle synthesis have been investigated and most of them were in accordance with our results. This means the choice of parameters in this work is reasonable.
Dynamic process of HGNBs at different temperatures
The geometric model of a HGNB was built in NanoMD as follows. First, an initial configuration was created with fcc structure lattices along the [100] crystallographic orientation, then the cubic was caved into a hollow structure with given values of the outer-diameter (R) and inner-diameter (r) as shown in Fig. 1. The momentum and coordinate space of each gold atom were recorded, from which the atom–atom interaction and potential energy of the HGNB were calculated. Then the HGNB system was relaxed until the potential energy reached equilibrium. Fig. 2 shows the potential energy of HGNBs with a 3 nm wall thickness and 14 nm diameter fluctuated at temperatures 4, 150, 300, 450 and 600 K respectively. The perfect single crystal structure gives a high potential energy at the beginning of all simulations (MD step = 0), and the initial energy is proportional to temperature. Then it drops sharply when relaxed (MD steps from 0 to 5 104) and keeps decreasing gradually (MD steps from 5 104 to 3 105), corresponding to the HGNB structure adjustment. Finally, the potential energy reaches equilibrium (MD steps after 3 105), which suggests a stable state for
ij
where E is the total internal energy, V(rij) is the pair interaction energy between an atom i and its neighboring atom j, j(rij) is
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Fig. 1
Visual images of initial state of hollow gold nanoball.
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Fig. 2 The evolution of potential energy of HGNBs with 3 nm wall thickness and 14 nm diameter, during 400 000 MD simulation steps at temperatures of 4, 150, 300, 450 and 600 K.
the HGNB. At 4 K, the potential energy of the HGNB fluctuates around the initial value during the whole simulation process, which indicates the original structure was kept perfectly. In this case the atomic rearrangement can hardly happen because at low temperatures the thermal energy of the gold particles is insufficient to overcome the energy barrier of slipping. In 150, 300, 450 and 600 K, the increasing thermal energy allows gold atoms to adjust their structure to achieve a lower potential energy state. It can be estimated that the energy of the system to should be between 0.475 V (4 K) to 0.46 V (150 K). At 600 K the potential energy reaches a plateau before achieving the equilibrium. This indicates that the system underwent a metastable state, which can be attributed to the thermal energy being so high that the gold particles move quickly enough to oppose the restructuring. The meta-stable state also relates to the size of the HGNBs. Fig. 3 shows potential energy evolution of HGNBs at 600 K with thickness of 3.5 nm and varying diameters of 24.5, 28 and 31.5 nm. It can be observed that a meta-stable plateau starts from approximate 9 104 MD steps for all the systems, suggesting that it is only a function of temperature but not diameter. However, the lengths of plateau are different. The meta-stable state ends at about 2.6 105 MD step for smallest HGNB system, which is four times longer than the HGNB of 28 nm diameter (plateau ends at 1.3 105 MD steps). For the HGNB of 31.5 nm, the semi-stable
Fig. 3 The evolution of potential energy of HGNBs with 3.5 nm wall thickness and different diameter values of 24.5, 28 and 31.5 nm at 600 K.
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plateau is hardly observed. This simulation result suggests that with the same wall thickness, the HGNBs of smaller diameter might undergo a longer meta-stable state. Corresponding to the potential energy curves in Fig. 2, Fig. 4 shows snapshots during the simulation process for a HGNB of 3 nm wall thickness and 14 nm diameter at 4, 150, 300, 450 and 600 K respectively. Those snapshots were captured as projections of gold atoms in the z plane. The first row snapshots are the initial structures at various temperatures. It can be observed that the atom arrangement gets less regular as the temperature increases because the atom vibration is fiercer. The structure re-arrangement at each temperature can be seen in the columns. At 4 K, the last snapshot, showing the projection after equilibrium, is similar to the initial one. It suggests that at 4 K the HGNB kept its initial structure during the whole relaxing process. At 150 K, as shown in the second column, the atoms of HGNB started to move and be reconstructed at [111] then the disorder spread to [110] and [100] gradually. Finally, the HGNB collapses to an irregular hollow structure, as shown in the last snapshot in column 2. More details regardinf this collapse process were given in the previous paper, and the cross-sectional images and animation gif files were attached in its supporting information.26 The result that the collapse started from [111] facet matches other experiments. Finbow et al. reported that at low temperatures nanoscale metal wires were found to slip over a [111] plane.45 The reason might be that for a close-packed structure such as fcc, the smallest Burgers vectors exist along the [110] direction, which makes it energetically favorable to reconstruct along the (111) slip planes. The similar collapse processes could be observed at 300 and 450 K in the third and forth column respectively. In 600 K, the HGNB collapsed isotropically, and maintained a semistructure in the process which corresponded to the meta-stable state in Fig. 2. And at the end, the HGNB it 600 K collapsed into a solid structure. In order to classify the HGNB systems, we defined three categories of sample stability, i.e. stable, meta-stable and unstable. A stable HGNB is one which can keep the initial potential energy
Fig. 4 Representative snapshots during the MD simulation process for the HGNBs with 3 nm wall thickness and 14 nm diameter at 4, 150, 300, 450 and 600 K.
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and atom structure, as does the HGNB of 3 nm wall thickness and 14 nm diameter at 4 K as shown in Fig. 2 and 4. A semistable HGNB is one in which the initial structure would collapse into an irregular hollow structure, such as HGNBs at 150, 300 and 450 K. For unstable HGNBs, these would collapse into a solid structure, as does the one at 600 K.
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3.2 Geometric thermal phase diagrams at different temperatures In our previous work,26,27 the aspect ratio is defined by p = R/d
(3)
where p is the aspect ratio, R is the outer radius, d is the wall thickness. Therefore, a HGNB would be described by d and p. As mentioned above, the geometric thermal phase diagram (GTPD) was developed by taking the aspect ratio (a.r.) as the y axis and wall-thickness (w.t.) as the x axis, which can be seen in Fig. 5. Each point in the graph stands for a HGNB system with a certain wall-thickness and aspect ratio. According to the three categories of stability discussed before, those points are shown as a square, circle or triangle representing stable, semi-stable and unstable HGNB systems respectively. In the previous work,26 at GTPD at 300 K was drawn, as shown in Fig. 5c. It has been well demonstrated that the thermal stability of HGNB depends on the a.r. and w.t. to a great extent. With the same aspect ratio, the HGNB with a larger w.t. is more stable. But with the same w.t., an increasing a.r. leads to a less stable HGNB. In this work, we are focused on how the temperature influences the stability of HGNB. The GTPDs at 4, 150, 450 and 600 K are shown in Fig. 5a, b, d and e respectively. In the 4 K GTPD (Fig. 5a), there is no unstable systems in the region studied, and the region of stable systems starts from a w.t. of 1.25 nm. The reason for the high stability is that the temperature is too low to provide enough kinetic energy for gold atoms to cross the energy barriers of atom slip and dislocation. At 150 K (Fig. 5b), the stable region starts from 2.0 nm w.t. and unstable systems can be found at 0.5–1.0 nm w.t. and 6–2 a.r. At 300 K (Fig. 5c), the stable system can only be found for w.t. larger than 2.5 nm and the unstable region enlarges to 1.5 nm. For 450 and 600 K (Fig. 5d and e), there is no stable system in the range studied, and the area of unstable HGNBs increases. To sum up, with increasing temperature, there are less stable systems and more unstable systems in the GTPDs. The areas of stable/semi-stable/unstable HGNB all move towards larger wall-thickness as the temperature rises. In other words, the stability of the HGNBs decreases. The decrease in stability is more obvious if we trace a certain size of HGNBs at increasing temperatures. For example, one HGNB with 8 a.r. and 1.5 nm w.t., was a stable system at 4 K which kept its initial structure. When the temperature increased to 150, 300 and 450 K, the HGNB underwent a structural re-arrangement and collapsed into an irregular hollow structure, appearing as a semi-stable system in the GTPD. At 600 K, the HGNB collapsed into a solid which indicated an unstable system.
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Fig. 5 The geometric thermal phase diagrams at temperatures of (a) 4 K, (b) 150 K, (c) 300 K, (d) 450 K and (e) 600 K. The boundary isometric line in b, c, d and e were fitted by eqn (4). The square, circle and triangle stand for stable, semi-stable and unstable HGNBs respectively.
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3.3 Boundaries and prediction of geometric thermal phase diagrams Limited by the computational capability, the simulations were carried out for the w.t. ranged from 0.5–3.0 nm and the a.r. from 2–18. However, by analysing the different areas and boundaries mathematically, we are able to predict the GTPDs beyond the computed range. An unstable HGNB is the one which collapses from its initial hollow structure gold sphere into an irregular solid ball. Imagine an ideal and ‘reversible’ collapse process wherein at each step the regular HGNB re-structures to another regular HGNB, until it reaches a regular solid sphere. In this situation, each rearranged HGNB can be presented as a point in the GTPD. All the points will compose a line segment. This line segment can be considered as part of a line, called an isometric line. All the points in this line share the same number of gold atoms. Since the gold atom number of a HGNB is kept constant, the isometric line can be described by eqn (4), as deduced in our previous work.26 d3[3p2 3p + 1] = C(T)
(4)
where C is a temperature-dependent constant (see below), p is the aspect ratio, and d is the wall thickness. In an ideal system, every unstable HGNB will collapse following an isometric line into a solid. In other words, if one point in the isometric line is found to be an unstable system, all the points in this line should also be unstable systems, sharing the same atom re-structure pathway. The unstable area is therefore composed of those lines. In a GTPD, HGNBs between unstable and semi-stable regions are close to the equilibrium, therefore these approach the ideal system if the step time is chosen small enough. In this case, the boundary line can be roughly fitted by eqn (4). Take the GTPD at 150 K for example, a point of 3 a.r. and 0.75 w.t. was chosen, which is between unstable points and semi-stable points, then the constant C can be calculated to be 8. An isometric line is therefore can be drawn in the GTPD, as displayed in Fig. 5b. By varying the point value in the area between unstable region and semi-stable region, different C values are obtained, as well as isometric lines. The one which fits the boundary most accurately should be chosen. In this way, the boundary line is achieved. The fitting lines of GTPDs in 150, 300, 450 and 600 K are shown in Fig. 5b–e, of which the C are found to be 8, 64, 120 and 1000 respectively. All the boundary fitting lines are drawn together in Fig. 6. It is apparent that with increasing temperature, the boundary fitting line moves toward higher a.r. and thicker w.t. Another thing that needs to be pointed out is that at lower temperatures, the accuracy and precision of the fitting of the boundary in the GTPD is better. In high temperatures, i.e. 600 K in Fig. 5e, it is hard to find an isometric line to fit to the boundary between the unstable region and semistable region. It is probably because the higher temperature drives the systems far away from the ideal equilibrium condition; therefore this method cannot be used. So far there is no proper theory that can be used to fit the boundary between
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Fig. 6 The fitting boundary isometric line in geometric thermal phase diagrams at 150, 300, 450 and 600 K with C equal to 8, 64, 120 and 1000 respectively.
semi-stable systems and stable systems. On the other hand, the boundaries between semi-stable systems and unstable systems, which are the focus of this work, are of more importance in practice. As reported experimentally, most of the hollow gold nanoballs (HGNBs) are semi-stable or unstable systems and the difference between them (having a hollow structure inside) is the major character of HGNBs and limits most of the applications such as drug delivery. As mentioned above, the value of constant C is a function of temperature. To investigate the relationship between them, more GTPDs in the temperature range of 150 to 700 K were simulated, drawn and fitted, where more values of the constant C were obtained. The reason to choose this temperature range is that there is hardly any unstable systems in the GTPDs at lower temperatures than 150 K (Fig. 5b), and it is difficult to accurately and precisely fit the GTPDs and find relationship between C and temperature at higher temperatures, as discussed previously. The plot of the logarithm of C against the inverse of temperature is displayed in Fig. 7, yielding two linear lines with different slopes. One is in the range from 150 to 450 K, corresponding to 0.0022 to 0.0067 in Fig. 7, and the other one is from 450 to 700 K, corresponding to 0.0022 to 0.0014. Therefore, in this range the constant C can be calculated if the temperature is known, and the boundary between unstable and
Fig. 7 The relationship between temperature and the constant in isometric line equation, 1/T vs. ln C.
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semi-stable areas in the GTPD can be deduced. Consequently the GTPD at this temperature can be predicted. The turning point of 450 K is of great interest. Although the physical meaning of this point is hard to fully understand, some explanation can be given. As seen from Fig. 2, the system energy at 450 K is close to the energy needed to slip. Therefore, the systems at higher temperatures may undergo a different structural re-arrangement from those at lower temperatures. In particular, at low temperature, the slipping along the (111) facet in the crystal hollow ball is the main mechanism of structural collapse. On the other hand, an isotropic structural collapse becomes the dominant mechanism at high temperatures. This might be the origin of the switch between two lines in Fig. 7. The GTPDs are in agreement to the experimental results from the literature.7,15–17 To the best of our knowledge, the HGNBs reported are all in the region of stable/semi-stable systems at 300 K (Fig. 5c). Dowgiallo et al. reported HGNBs with aspect ratios from 1.6 to 4 and wall thicknesses from 3 to 11 nm.18 This is in the stable region of the GTPD at 300 K (Fig. 5c), showing a high stability at room temperature. However, this HGNB should not be heated to over 450 K in experiment since the HGNB may be in the semi-stable region (Fig. 5d). And if the temperature is higher than 600 K, the HGNB may collapse into a solid sphere (Fig. 5e).
4 Conclusion In this work, we investigated how the temperature and the geometric features (wall thickness and aspect ratio) would affect behaviours of HGNBs. The increasing temperature leads to higher average atomic kinetic energy, then less stability and at last faster collapse processes. However, when the temperature is high enough, a semi-stable stage appears, corresponding to a homogeneous collapse process. The geometric thermal phase diagrams, which indicates the thermal dynamic stability of a HGNB with a certain wall-thickness and aspect ratio, was developed over a wider temperature range from 4 to 600 K. As temperature increased, the unstable region in the GTPD is enlarged. The boundaries between the unstable and semi-stable regions in GTPDs over the temperature range from 4 to 600 K were fitted. The relationship between the boundary line and temperature enables us to propose probable GTPDs at other temperatures, and predict the thermal stability of HGNBs at different temperatures.
Acknowledgements This work was funded by National Key Technology R&D Program of China 2012BAF03B05 and the National Natural Science Foundation of China (NSFC) No. 21121091, 21273113, and 51071084.
References 1 H. C. Zeng, J. Mater. Chem., 2006, 16, 649. 2 J. Meier, J. Schiotz, P. Liu, J. K. Norskov and U. Stimming, Chem. Phys. Lett., 2004, 390, 440.
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3 S. Link and M. A. El-Sayed, Int. Rev. Phys. Chem., 2000, 19, 409. 4 Y. Ding, F. R. Fan, Z. Q. Tian and Z. L. Wang, J. Am. Chem. Soc., 2010, 132, 12480. 5 V. Uzayisenga, X. D. Lin, L. M. Li, J. R. Anema, Z. L. Yang, Y. F. Huang, H. X. Lin, S. B. Li, J. F. Li and Z. Q. Tian, Langmuir, 2012, 28, 9140. 6 Y. G. Sun, B. Gates, B. Mayers and Y. N. Xia, Nano Lett., 2002, 2, 165. 7 Y. G. Sun, B. Mayers and Y. N. Xia, Adv. Mater., 2003, 15, 641. 8 P. R. Selvakannan and M. Sastry, Chem. Commun., 2005, 1684. 9 X. Sun, C. Zhai and X. Y. Wang, Bioprocess Biosyst. Eng., 2013, 36, 273. 10 W. Zhu, T. Chen, X. Ma, H. Ma and S. Chen, Colloids Surf., B, 2013, 111, 321. 11 M. Chandra, A. M. Dowgiallo and K. L. Knappenberger, J. Phys. Chem. C, 2010, 114, 19971. 12 A. M. Schwartzberg, T. Y. Olson, C. E. Talley and J. Z. Zhang, J. Phys. Chem. B, 2006, 110, 19935. 13 J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, D. Y. Wu, B. Ren, Z. L. Wang and Z. Q. Tian, Nature, 2010, 464, 392. 14 M. Chandra, A. M. Dowgiallo and K. L. Knappenberger, J. Am. Chem. Soc., 2010, 132, 15782. 15 Y. G. Sun and Y. N. Xia, J. Am. Chem. Soc., 2004, 126, 3892. 16 K. L. Knappenberger, A. M. Schwartzberg, A. M. Dowgiallo and C. A. Lowman, J. Am. Chem. Soc., 2009, 131, 13892. 17 S. Preciado-Flores, D. C. Wang, D. A. Wheeler, R. Newhouse, J. K. Hensel, A. Schwartzberg, L. H. Wang, J. J. Zhu, M. Barboza-Flores and J. Z. Zhang, J. Mater. Chem., 2011, 21, 2344. 18 A. M. Dowgiallo, A. M. Schwartzberg and K. L. Knappenberger, Nano Lett., 2011, 11, 3258. 19 C. W. Huang, J. C. Jiang, M. Y. Lu, L. Sun, E. I. Meletis and Y. W. Hao, Nano Lett., 2009, 9, 4297. 20 S. N. Abdollahi, M. Naderi and G. Amoabediny, Colloids Surf., A, 2013, 436, 1069. 21 J. H. Lin, W. D. He, S. Vilayurganapathy, S. J. Peppernick, B. Wang, S. Palepu, M. Remec, W. P. Hess, A. B. Hmelo, S. T. Pantelides and J. H. Dickerson, ACS Appl. Mater. Interfaces, 2013, 5, 11590. 22 B. S. Gutrath, M. F. Beckmann, A. Buchkremer, T. Eckert, J. Timper, A. Leifert, W. Richtering, G. Schmitz and U. Simon, Nanotechnology, 2012, 23, 225707. 23 G. Schneider and G. Decher, Nano Lett., 2004, 4, 1833. 24 Y. D. Yin, C. Erdonmez, S. Aloni and A. P. Alivisatos, J. Am. Chem. Soc., 2006, 128, 12671. 25 T. Chookajorn, H. A. Murdoch and C. A. Schuh, Science, 2012, 337, 951. 26 L. Y. Jiang, X. Yin, J. W. Zhao, H. M. Liu, Y. H. Liu, F. Y. Wang, J. J. Zhu, F. Boey and H. Zhang, J. Phys. Chem. C, 2009, 113, 20193. 27 L. Y. Jiang, X. Yin and J. W. Zhao, Chin. J. Inorg. Chem., 2009, 25, 176. 28 A. Rahman, Phys. Rev., 1964, 136, A405.
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View Article Online
PCCP
29 30 31 32
Published on 16 January 2014. Downloaded by Arizona State University on 30/10/2014 18:13:43.
33 34 35 36 37 38
B. J. Alder and T. E. Wainwright, J. Chem. Phys., 1959, 31, 459. S. Nose, Mol. Phys., 1984, 52, 255. S. Nose, J. Chem. Phys., 1984, 81, 511. F. Y. Wang, Y. H. Liu, T. M. Zhu, Y. J. Gao and J. W. Zhao, Nanoscale, 2010, 2, 2818. A. D. Becke, J. Chem. Phys., 1993, 98, 5648. A. D. Becke, Phys. Rev. A: At., Mol., Opt. Phys., 1988, 38, 3098. R. A. Johnson, Phys. Rev. B: Condens. Matter Mater. Phys., 1988, 37, 3924. R. A. Johnson, Phys. Rev. B: Condens. Matter Mater. Phys., 1989, 39, 12554. M. S. Daw, S. M. Foiles and M. I. Baskes, Mater. Sci. Rep., 1993, 9, 251. M. S. Daw and M. I. Baskes, Phys. Rev. B: Condens. Matter Mater. Phys., 1984, 29, 6443.
This journal is © the Owner Societies 2014
Paper
39 M. S. Daw and M. I. Baskes, Phys. Rev. B: Condens. Matter Mater. Phys., 1984, 29, 6443. 40 F. Y. Wang, T. N. Chen, T. M. Zhu, Y. J. Gao and J. W. Zhao, J. Appl. Phys., 2011, 110, 1082. 41 Y. J. Gao, Y. Q. Fu, W. Sun, Y. L. Sun, H. B. Wang, F. Y. Wang and J. W. Zhao, Comput. Mater. Sci., 2012, 55, 322. 42 D. X. Wang, J. W. Zhao, S. Hu, X. Yin, S. Liang, Y. H. Liu and S. Y. Deng, Nano Lett., 2007, 1208. 43 Y. H. Liu, J. W. Zhao and F. Y. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 115417. 44 J. W. Zhao, K. Murakoshi, X. Yin, M. Kiguchi, Y. Guo, N. Wang, S. Liang and H. Liu, J. Phys. Chem., 2008, 112, 20088. 45 G. M. Finbow, R. M. LyndenBell and I. R. McDonald, Mol. Phys., 1997, 92, 705.
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