J. Phys. Chem. B 2005, 109, 9703-9711
9703
Surface Diffusion and Coalescence of Mobile Metal Nanoparticles M. Jose´ -Yacama´ n,†,‡ C. Gutierrez-Wing,†,‡,§ M. Miki,†,| D.-Q. Yang,⊥ K. N. Piyakis,⊥ and E. Sacher*,⊥ Center for Nano and Molecular Technology, Texas Materials Institute, Department of Chemical Engineering, The UniVersity of Texas at Austin, Austin, Texas 78712, Instituto Nacional de InVestigaciones Nucleares, Carr. Me´ xico-Toluca, Km 36.5, Ocoyoacac Edo. de Me´ xico 52045, Me´ xico, Centro de InVestigacion en Materiales AVanzados, Miguel de CerVantes #120, Chihuahua, Chih. C. P. 31109, Me´ xico, and Regroupement Que´ be´ cois de Mate´ riaux de Pointe, De´ partement de Ge´ nie Physique, EÄ cole Polytechnique, C.P. 6079, Succursale Centre-Ville, Montre´ al, Que´ bec H3C 3A7, Canada. ReceiVed: February 23, 2005; In Final Form: March 17, 2005
The diffusion and coalescence of metal nanoparticles play important roles in many phenomena. Here, we offer a new integrated overview of the main factors that control the nanoparticle coalescence process. Three factors are considered in our description of the coalescence process: nanoparticle diffusion across the surface, their physical and thermodynamic properties, and the mechanism of coalescence. We demonstrate that the liquidlike properties of the surface layers of the nanoparticles play an essential role in this process. We present experimental evidence for our opinion, based on the high-resolution electron microscopic analysis of several different types of nanoparticles.
Introduction It is evident that a technology may be more easily exploited and optimized when the science permitting its exploitation is understood. For this reason, much effort is presently being expended in the area of nanoscience, probing what may ultimately become nanotechnology. There has been a vast increase of interest in nanoparticles, driven by their size and their unique physical and chemical properties. These properties are already being exploited in commercial areas as diverse as chemical catalysis, biological sensors, and electronics. One area of knowledge necessary to their use is their dimensional stability on surfaces. This area has been the subject of several recent reviews1-4 as well as several articles containing information of a more general interest.5-12 A most important aspect of metal nanoparticle stability on surfaces is their capacity to diffuse across the surface and to coalesce on contact. This leads to nanoscale dimensional changes, with accompanying changes in the desirable physical and chemical properties, possibly leading to device failure. For nanoparticles on surfaces, there appear to be two limiting cases of the kinetics of dimensional changes: (a) Coalescence, in which the nanoparticles adhere poorly to the surface, permitting them to diffuse across it and to coalesce on contact; (b) Ostwald ripening, in which the nanoparticle adheres strongly to the surface, making atomic transfer between nanoparticles more favorable. * Author to whom correspondence should be addressed. Phone: (514) 340-4711 ext 4858. Fax: (514) 340-3218. E-mail:
[email protected]. † Center for Nano and Molecular Technology, Texas Materials Institute, The University of Texas at Austin. ‡ Department of Chemical Engineering, The University of Texas at Austin. § Instituto Nacional de Investigaciones Nucleares. | Centro de Investigacion en Materiales Avanzados, Miguel de Cervantes #120. ⊥ Regroupement Que ´ be´cois de Mate´riaux de Pointe, De´partement de Ge´nie Physique, EÄ cole Polytechnique.
Both processes generally follow the von Smoluchowski kinetic rate equation,13 d ∝ t-R, where d is the nanoparticle dimension, t is the time, and R is a constant. They differ in the magnitude of R, with R decreasing with increasing particlesubstrate interfacial adhesion.14 Here, we consider case a above; while coalescence has also been found for nonmetals, we limit our discussion to metal nanoparticles. We begin with basic considerations that we believe to be important in the coalescence process, which come from our own work and that of others. These considerations develop our point of view of the coalescence process, which we substantiate with new experimental data on metal nanoparticles. Basic Considerations for the Coalescence Process We present here our view of the various factors contributing to the coalescence process. The points that we consider, many of them stated for the first time, are those that we have found important in understanding the coalescence process and in formulating our view of why coalescence takes place, how this process depends on nanoparticle properties, and the overall mechanics of the coalescence process. 1. Factors Influencing Nanoparticle Diffusion near Room Temperature. Diffusion along the substrate is due to weak nanoparticle interfacial adhesion. Since, in the present case, the activation energy for Brownian motion is less than that for Ostwald ripening, Brownian motion is the preferred diffusion mechanism. While both types of behavior follow the equation first proposed by von Smoluchowski,13 d ∝ t-R, the value of R is strongly affected by interfacial adhesion.15,16 The energy necessary for this process to occur is taken from the thermal background. While nanoparticle diffusion across a surface may be more rapid, e.g., in the high-resolution electron microscope (HREM),3 where it occurs under the influence of the high-intensity electron beam, it also occurs at room temperature and below, albeit more slowly,1-4 because it must obtain its energy from the thermal background. We take it for
10.1021/jp0509459 CCC: $30.25 © 2005 American Chemical Society Published on Web 04/15/2005
9704 J. Phys. Chem. B, Vol. 109, No. 19, 2005 granted that diffusion is a thermally activated kinetic process that, in a HREM, is enhanced by the energy of the beam. Almost two decades ago, it was found that Au nanoparticles undergo structural fluctuations in the beam of a HREM.17,18 Several attempts were made to explain these results,3 assuming that the structural fluctuations were caused by the energy transferred from the beam. When these processes were followed for extended periods of time, no loss of material was noted from the nanoparticles being observed. We believe that such Brownian motion occurs through particle shape oscillations, provoked by slip-plane dislocations that occur with time, causing small changes in the center of gravity of the nanoparticle that, together with the weak adhesion to the substrate, lead to slight positional shifts with each oscillation. Such dislocations occur through the discrete simultaneous motion of a few atoms, which takes far less energy than does mass motion along a slip plane. Such motions may be described by the classic Frank-Read mechanism19 for slip. The result of the slip is a distortion, viewed as a structural fluctuation, which changes the center of gravity of the nanoparticle on the substrate, thus changing its point of contact. This appears, over time, as small, random displacements across the surface: surface diffusion. Such dislocation and slippage processes are activated,20 meaning that they may be pressure- as well as temperature-dependent. This will be considered in the next section. 2. Nanoparticle Properties and Their Influence on Coalescence. The melting points of nanoparticles have been found to be substantially lower than the bulk value, increasing with nanoparticle diameter until they become asymptotic with that of the bulk.21-23 The first authors to show this in Au nanoparticles21 offered two phenomenological models for this behavior, the second of which is based on a solid particle embedded in a thin molten layer (the first is based on the simultaneous existence of solid, liquid, and vapor phases and has no relation to the experimental facts for the metal nanoparticles considered here). Excellent agreement with experiment was obtained for this second model, when the liquid-layer thickness, δ, was 0.62 nm (i.e., 1.8 times the van der Waals diameter) and the interfacial tension, γsl, was 266 mN m-1. These latter values agree satisfactorily with values given in their Table 1, previously determined experimentally.24 It is interesting to speculate on the possible origin of such a liquid surface layer. While a recent molecular dynamics (MD) simulation on a totally crystalline Au nanoparticle showed that such a layer formed as a premelting phenomenon,25 it is interesting to note the recent success of the bond order-bond length-bond strength correlation of Sun;22,23 this correlation is based on the assertion of Goldschmidt,26 which states that the ionic radius of an atom contracts as its coordination number is reduced and crystallinity is lost at the surface. This approach was used by Sun22,23 to correlate the melting points of several materials with their nanoparticle diameters. The stabilizing effect of isotropic interactions does not exist at the surface, and this surface asymmetry is the reason for the existence of a surface energy, which is the impetus for coalescence, if the barriers to such coalescence27 are not too high. The driving force for the coalescence of two nanoparticles is the surface energy reduction experienced because the surface area of the new nanoparticle is less than that of the sum of the surface areas of the original two nanoparticles. However, this driving force decreases with increasing nanoparticle size. To see this, we consider two identical spherical nanoparticles coalescing. For the volume, V
Jose´-Yacama´n et al.
Vbefore ) (2‚4π/3)r3before
(1)
Vafter ) (4π/3)r3after
(2)
and
) (2‚4π/3)r3before
(3)
For the surface area, Ω
Ωbefore ) 2‚4πr2before
(4)
Ωafter ) 4πr2after
(5)
and
) 4π‚22/3r2before
(6)
) 2-1/3Ωbefore
(7)
Thus, for a constant volume
∆Ω/V ) (Ωbefore - Ωafter)/[(2‚4π/3)r3before] ) 3(1 - 2-1/3)/rbefore ) 0.62/rbefore
(8) (9) (10)
Therefore, as rbefore increases, ∆Ω/V decreases. That is to say that the Helmholz surface energy per unit area, a (A/Ω), decreases as r increases. Although the percentage of the surface area decrease is independent of the original nanoparticle size, the percent decrease per constant Volume of the nanoparticles decreases with coalescence, continually reducing this effect as a driving force for coalescence. That is to say that, for any constant mass of nanoparticles, the tendency to coalesce decreases with the extent of coalescence. There is another reason for the extent of diffusion to decrease with increasing nanoparticle size. This is a consequence of the diffusion coefficient, D, being inversely related to the nanoparticle dimension, d, as recently shown:15 D ∝ d-(1-3R/R). For sufficiently small nanoparticle sizes, the pressure within the nanoparticle may be elevated by its high radius of curvature. This increased pressure permits plastic deformation through the slip of edge defects, either within the nanoparticle or at the facet joints. The pressure within a spherical particle may be calculated by the Young-Laplace equation and approximates that within real nanoparticles, which are not truly spherical. The YoungLaplace equation relates the increase in pressure above atmosphere, ∆P, to the radius of curvature, r; ∆P ) 2γs/r, where γs is the surface tension. For gold near its melting point,28 γs ≈ 1400 mN m-2, leading to the values in Table 1. Thus, the crystalline metal beneath the liquidlike surface layer can, depending on the nanoparticle radius, be under tremendous pressure. This has been shown to be true experimentally,29 as seen by using scanning high-energy electron diffraction; indeed, these authors discuss the Young-Laplace equation from the point of view of Chaˆtelain,30,31 who calculated ∆P for several equilibrium crystalline shapes. 3. Nanoparticle Coalescence Process Mechanism. The coalescence process begins with contact and initial fusion, followed by the orientational alignment of coalescing planes at the interface between the particles, on contact. For this to happen, the surface atoms must be mobile (see subsection 2, above). Reorganization at the coalescing particle interface was predicted by the molecular dynamics simulations of Zhu and Averback,32 who found that it served to reduce the energy at the grain boundary (see subsection 1, above). The stress
Coalescence of Metal Nanoparticles
J. Phys. Chem. B, Vol. 109, No. 19, 2005 9705
TABLE 1: Pressure Increase in a Sphere as a Function of Its Radius of Curvature radius of curvature (nm)
∆P (MPa)
∆P (atm)
0.5 5 50 500 5000
56 5.6 0.56 0.056 0.0056
5527 553 55 5.5 0.6
developed at the boundary exceeded the theoretical strength of the particles, which coalesced through the glide of dislocations. Their model has recently been questioned by an MD simulation,25 although both are in qualitative agreement with our experimental results. In a series of papers, Lehtinen33-35and Zachariah33,34,36 discussed the coalescence process. Their work indicates that, as mentioned in subsection 2, above, the loss in surface energy due to the loss in surface area is the driving force for coalescence. The heat created at the coalescing interface melts more crystalline material (see subsection 2, above), with heat transferred to the volumes of the coalescing particles as well as to the surroundings. With the increase in the nanoparticle melting point with increasing size and the conduction of heat to a larger mass, the ability to melt the whole of the coalescing nanoparticles becomes more and more difficult. The loss of heat from the coalescing boundary ultimately becomes too great to further melt the coalescing nanoparticles, leading to only partial coalescence. Such partially coalesced particles are invariably found only for larger nanoparticle masses. There may be concern as to whether the pressure acting on the coalescing nanoparticles may influence their rate of coalescence.This is because the enthalpy of activation of the coalescence process is ∆Hq ) ∆Eq + P∆Vq, where ∆Eq is the activation energy and ∆Vq is the volume of activation; thus, (∂
ln k/∂P)T ) -∆Vq/RT, where k is the rate constant for the coalescence process, R is the gas constant, and T is the absolute temperature. Therefore, a negative value of ∆Vq will cause the rate constant to decrease with decreasing pressure, and a positive value will cause the rate constant to increase. A small number of ∆Vq measurements are found in the literature, covering such phenomena as deformation in alloys,37,38 rhombohedral twinning in saphhire,39 the transition between 4-coordination and 6-coordination in CdSe,40 and atomic diffusion in amorphous alloys.41-43 Of these, only the latter three remotely correspond to the case of metal coalescence. All fall into the range of 8-20 Å3, with an uncertainty roughly that of the value determined;43,44 given this uncertainty and the generally mild pressures in Table 1 (except for extremely small nanoparticles), it would appear that pressure has a limited influence on coalescence, confined to extremely small (
