J. Phys. Chem. C 2008, 112, 14731–14736
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ARTICLES Free Energy Change of Aggregation of Nanoparticles Dino Spagnoli,†,‡ Jillian F. Banfield,† and Stephen C. Parker*,‡ Department of Earth and Planetary Science, UniVersity of California, Berkeley, California 94720, and Department of Chemistry, UniVersity of Bath, Bath, BA2 7AY, U.K. ReceiVed: NoVember 7, 2007
We used molecular dynamics simulations to calculate the free energy change due to aggregation of MgO nanoparticles in vacuum and examine its dependence on particle size and interparticle orientation. High quality interatomic potentials with a proven track record for simulation of surface and bulk properties, including a representation of electronic polarizability were deployed. The calculations generally predict a free energy barrier to aggregation. However, the free energy barrier to aggregation can be removed by allowing the particles to approach in a crystallographically aligned manner. This implies that aggregation may not be an energetic imperative, but can occur as the result of fluctuations in orientation. Lowering of nanoparticle energy by change in orientation may drive crystal growth via oriented aggregation. Introduction Aggregation is a fundamental process important in crystal growth and formation and destabilization of colloids. Consequently, understanding of the aggregation process is of technological, geochemical and biological importance. Of particular interest to us are the steps that lead to the conversion of two or more nanoparticles into a single crystal, a process generally described as oriented aggregation (OA).1,2 As for atom-by-atom growth, OA is driven by surface energy reduction, an outcome achieved by complete elimination of surfaces at particle-particle interfaces. OA-based crystal growth has primarily been studied by imaging the resulting nanoparticles3,4 and kinetic analysis.5 Missing from current understanding is quantitative information about the driving forces that lead to formation of aligned particles, a key step in OA-based crystal growth. Experimentally, quantitative measurement of forces between particles within aggregates is difficult. One technique employs a Hamaker constant6 to estimate the dispersion forces between macroscopic objects from spectroscopic measurements.7 However, the Hamaker constant is based on number densities and polarizabilities of atoms of interacting bodies and ignores the effects of the surrounding medium. A more sophisticated approach has been used by Bergstorm,8 who used Lifshitz theory9 to calculate the Hamaker constant as function of the chemical surroundings. Other studies addressed the attractive forces in cement suspensions,10 using rigorous continuum theory to calculate the dispersion forces with a set of well-characterized dielectric properties. All are based on the premise that for neutral particles the energy of interaction is always attractive. Atomistic simulations have been used to describe the structure, stability and properties of bulk minerals,11,12 mineral * To whom correspondence should be addressed. Phone: +44 (0) 1225 386505. Fax: +44 (0) 1225 386231. E-mail:
[email protected]. † University of California. ‡ University of Bath.
surfaces,13,14 and discrete nanoparticles.15-17 The structures and stabilities of single nanoparticles have been calculated and it has been demonstrated that these are dependent upon environmental conditions.18-20 Huang et al. were one of the first to show that molecular dynamics (MD) can be used to describe aggregation of nanoparticles and predict experimentally observed phase transitions driven by aggregation.21 In this paper we use MD to calculate the free energy change of aggregation for pairs of particles as an alternative method for obtaining a quantitative estimate of the dispersion forces. Methodology. The interatomic potentials used in this work13,22-24 have been used in a number of studies of surfaces and grain boundaries25-29 and in each case the result agrees with the available experimental data. We include the shell model of Dick and Overhauser,30 which accounts for electronic polarization of ions. In the MD simulations the motion of the shells was treated using the adiabatic approach as described by Mitchell and Fincham,31 where by the shells are assigned a small fictional mass (0.2 au). All MD simulations were performed at 300 K in the NVE ensemble (constant number of particles, constant volume and constant energy) using the computer code DL_POLY32 with no periodic boundary conditions. We placed two charge neutral, identical nanocrystals at center-to-center separations of 7-30 Å. The cutoff (60 Å) used in all simulations was large enough to include the longest particle to particle separation. To calculate the free energy change, we use potential of mean force (PMF) calculation. This is a generalization of interparticle interaction energy along a reaction coordinate, the separation distance. If a number of simulations are conducted at different separation distances, the mean force may be plotted as a function of distance and the function integrated to obtain the free energy of the overall process. In each calculation the step in distance for the thermodynamic integration is 1 Å. This technique has been used in the past to measure the intermolecular forces between two gold passivated nanoparticles.33 In our simulations, the reaction
10.1021/jp804966c CCC: $40.75 2008 American Chemical Society Published on Web 08/30/2008
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Figure 1. Free energy change due to aggregation as a function of distance for a 1 nm MgO nanoparticle. Note the existence of an energy barrier to aggregation that is negligible until the particles can touch (closer than 15 Å).
coordinate is defined as the distance between the central cations in the two nanocrystals. We constructed two 1.0 nm diameter nanoparticles of the mineral periclase, MgO, with a morphology obtained from a Wulff construction.34 The equilibrium morphology for MgO was a cube bounded by the stable (100) surface. The cubic morphology of nanocrystalline MgO has been observed experimentally and characterized by scanning electron microscopy (SEM) and transmission electron microscopy (TEM).35 Two {100} 1.0 nm MgO nanoparticles were placed in a range of 7-30 Å apart and twenty independent MD simulations (0.2 fs time step) performed with no periodic boundary conditions. Each simulation was run for 1 ns with the first 200 ps used for equilibration and the following 800 ps used for averaging the PMF. Results and Discussion Although the distances between the centers of the nanoparticles are fixed, nanoparticles are free to rotate around their own axis and around each other. We observe a large decrease in free energy as nanoparticle separation decreases from 15 to 7 Å (Figure 1). When separated by 15 Å, the corners of nanoparticles are able to touch, and do so periodically during the simulation. It is interesting to note that the free energy decrease with decreasing separation distance is due to alignment of the particles. As the nanoparticles approach, they bond at a corner, then edge, and then orient with {100} faces parallel, ultimately forming an elongated single nanocrystal (Figure 2). The calculated free energy change when the particles join is -1783.5 kJ/mol (Figure 1). As the faces of MgO nanoparticle are comprised of 8 of each type of atom, 16 new magnesium-oxygen bonds form. Thus, the predicted bond energy is -111 kJ/mol. This compares favorably to the potential energy of a magnesium-oxygen bond in the bulk, -108.1 kJ/mol. A surprising result was the maximum in free energy at separation distances of ∼15 Å, just before the nanoparticles are within atom-atom contact, suggesting a kinetic barrier to aggregation in vacuum. To test whether this was the position of the free energy maximum, ten further calculations were performed with a step of 0.1 Å between 15 and 16 Å (Figure 1). The free energy barrier maximum occurs just
Figure 2. Final configuration of two 1 nm MgO nanoparticles at a center-to-center distance of (i) 15, (ii) 12, and (iii) 9 Å. The three energy measurements for these cases are included in Figure 1.
before the particles can touch at 16 Å. Previous MD studies,19,25 show that nanoparticles in vacuum aggregate without energy input. Free-energy barriers to aggregation are evident and discussed in colloidal science, and are generally associated with Coulombic repulsions between charged particles in solution. In order to examine the origin of the energy barrier to aggregation in vacuum and determine its dependence on nanoparticle composition and size, the calculations described above were conducted for two {101j 4} 1.6 nm calcite nanoparticles and two 1.7 nm {100} MgO nanoparticles (Figure 3). The same energy barrier predicted for the 1.0 nm diameter MgO nanoparticles preceding touching of nanoparticles was observed for the larger MgO nanoparticles, as well as for calcite nanoparticles. Another difference between MgO and CaCO3 nanoparticles at these sizes is that calcite is thermodynamically more stable as an amorphous particle. We have attempted to identify the reasons for the freeenergy maximum by considering the separate contributions toward the free-energy. In Figure 4 we have extracted the thermally averaged configurational (the total interaction energy) and Coulombic energy components as a function of distance. In the region where the nanoparticles are not attached (16-26 Å) the configurational and Coulombic energy terms, show an increase at 16 Å relative to two isolated nanoparticles. As the nanoparticles attach at corners,
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Figure 3. Free energy change of aggregation as a function of distance for a 1.7 nm diameter MgO (dashed line) and a 1.6 nm diameter calcite nanoparticle (solid line). The results indicate that the energy barrier for aggregation of 1.7 nm MgO occurs at the same separation distance as for 1.0 nm MgO (Figure 1) and that calcite also exhibits an energy barrier to aggregation.
Figure 4. Interaction energy as a function of distance for the 1.0 nm MgO system broken down into configurational energy and Coulombic energy relative to two isolated nanoparticles.
edges and then steps the configurational energy then reduces, as to be expected because the surfaces of the nanoparticles are being decreased. This implies that the free energy maximum arises from the net repulsion between the particles. However, if there is on average a net repulsion, why should the particles want to attach as found in previous MD studies.19,25 Thus we examined the interaction of particles at specific orientations. We investigated the magnitude of the interactions by fixing the nanoparticles so that no rotation could occur as they approached one another. The two different sized MgO nanoparticles were set up in the same way as previous calculations, however, in addition to the PMF constraint, cations on the opposite surfaces of each nanoparticle were fixed so that the nanoparticles were unable to rotate. Three different orientations were considered; a completely aligned orientation, where the {100} faces are parallel to each other (Figure 5i), [100] parallel to [110] (Figure 5ii), and a random orientation (Figure 5iii) equivalent to a mixed twist and tilt boundary. The free energy profile of both the 1.0 (Figure 5B) and 1.7 nm (Figure 5C) MgO nanoparticles shows the
same trend. If the nanoparticles approach for a completely aligned attachment along the {100} face, the free energy curve is always negative (i.e., no free energy barrier), indicating that this is the preferred aggregation pathway. Nanoparticles that approach in misaligned (ii) and random (iii) orientations show a positive free energy profile, indicating that this is a highly undesirable process. At very near distances for the unaligned attachment of the 1.0 nm MgO nanoparticle, the free energy drops to -126.9 kJ/mol. In this case, attraction is so strong for aligned attachment that the nanoparticles reconstruct around the fixed atoms so as to orient at the surface (Figure 5B). Clearly the orientation in which the nanoparticles approach each other will determine if there is a repulsive or attractive free energy change. We explored the nature of the energy barrier by performing a series of static lattice simulations, where the distance was fixed but, before relaxation was allowed, one of the nanoparticles was rotated in 0.1° intervals (Figure 6). The distance was fixed at 16 Å, where the maximum free energy was observed in Figure 1. We show the interaction energy allowing for full atom relaxation
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Figure 5. (A) Free energy change for aggregation was calculated as the force needed to maintain the separation of two MgO nanoparticles that are fixed in (i) crystallographically aligned, (ii) [100] parallel to [110] and (iii) randomly aligned orientations. (B and C) The results for 1 and 1.7 nm (diameter measured along the body diagonal) MgO nanoparticles, respectively. Arrows indicate the center-to-center separation distances at which the particles touch.
(Figure 6A), except for the central atoms. The results show a strongly attractive interaction, however there are fluctuations in interaction energy depending on the angle at which the nanoparticle is rotated. In subsequent simulations we held the atoms fixed at their average sites, although the shells were allowed to relax (Figure 6B and C). Shell relaxation is on a faster time scale than atom vibration so we included this relaxation to see its overall effect on the interaction energy as a function of rotation. The calculation was performed twice. First, the nanoparticles were in an initial configuration parallel to each other (Figure 6B) and second they had an arbitrary initial orientation (Figure 6C). In both, there are attractive and repulsive interactions because in practice there are many possible orientations and atom locations. Clearly, the PMF results show that on average the net interaction is repulsive
and it is only via the fluctuations in rotation that the system can achieve a strong preferred orientation, as we have observed both in previous simulations19,25 and experimental studies.1,2 Indeed, returning to the original PMF calculations we noted that the particles continued to rotate freely until the particles touched. At the longest separation, where the nanoparticles are 26 Å apart in the 1.0 nm diameter MgO system, each nanoparticle rotates around it own axis once every 3.2 ps. Therefore over the course of the 1 ns simulation the nanoparticle rotates 312 times. Moreover the nanoparticles rotate around each other, on average, once every 87 ps. As previously stated, the free energy maximum for the 1.0 nm MgO system occurs at a separation of 16 Å apart and at this distance a single nanoparticle rotates around it is own axis once every 12 ps and rotates around the other nanoparticle
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Figure 6. Interaction energy between two 1.6 nm particles as one of the nanoparticle is rotated by 0.1° intervals around the normal between the two centers. In each case the central atoms are held fixed. (A) After each rotation all atoms are allowed to relax, (B) after each rotation only the shells are allowed to relax, (C) after each rotation only the shells are allowed to relax, however, the nanoparticle is rotated in a different orientation.
once every 98 ps. Therefore during the simulation the nanoparticles are not getting caught in a local minimum and are sampling the different preferred orientiations prior to bonding. Conclusion The results of this work show that when uncharged freely rotating cubic MgO particles approach each other in vacuum there is, on average, a free energy barrier to attachment. We attribute the barrier to significant repulsions between the atoms in the two particles when they are not aligned, although loss of rotational entropy may also be a factor. In previous MD simulations,19,25 where nanoparticles were able to freely rotate, no energy barrier was observed. The driving force for aggregation can therefore be attributed to nanoparticle rotation into crystallographic alignment. The size of the energy barrier to misaligned aggregation, compared to the free energy of the aligned pathway, may explain why nanoparticles attach in crystallographically specific orientations, an important step in grain growth under some conditions. As the particles approach they continue to spin, which means that on average the particles spend more time experiencing a repulsive potential and it is only when they are actually close enough to bond that they then attach. The particles are sampling different possible routes until the particles choose the most favorable orientation. The particles do not stop their rotation to follow the minimum energy path because of the mobility of the particles. This may be attributed to the more complex interaction energy generated by polarizable atoms, which gives rise to a more complex energy surface, which the particles experience before actually bonding. In addition, unlike phenomenological models that suggest that neutral particles are always energetically attractive, the atomistic simulations suggest that on average they are repulsive, although preferred orientations allow for attachment. MD simulations also provide an approach for estimating the magnitude of the driving force for OA for different materials. This method has been used to describe the free energy change
as nanoparticles come together with particle emphasis on the cubic morphology of MgO. We consider this method could be used to study aggregation of different materials and morphologies. The results also imply that the preferred orientation is highly dependent on short ranged interactions, which can be addressed well by atom-level techniques and hence this offers an alternative route for identifying ways of modifying the process. The next step is to investigate how energies are altered by the presence of an aqueous solution between the aggregating nanoparticles. Acknowledgment. This work was supported with the computer resources provided by the MOTT2 facility (EPSRC Grant No. GR/S84415/01) run by the CCLRC e-Science Centre. Funding for a component of the research was supported by the Director, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The authors thank Dr. Glenn A. Waychunas for access to the Geochemistry computer cluster at the Lawrence Berkeley National Laboratory. We thank Dr. David J. Cooke and Dr. Arnaud Marmier for their useful discussions. References and Notes (1) Penn, R. L.; Banfield, J. F. Science 1998, 281, 969. (2) Penn, R. L.; Banfield, J. F. Am. Mineral. 1998, 83, 1077. (3) Guyodo, Y.; Mostrom, A.; Penn, R. L.; Banerjee, S. K. Geophys. Res. Lett. 2003, 30. (4) Calderone, V. R.; Testino, A.; Buscaglia, M. T.; Bassoli, M.; Bottino, C.; Viviani, M.; Buscaglia, V.; Nanni, P. Chem. Mater. 2006, 18, 1627. (5) Davis, T. M.; Drews, T. O.; Ramanan, H.; He, C.; Dong, J. S.; Schnablegger, H.; Katsoulakis, M. A.; Kokkoli, E.; McCormick, A. V.; Penn, R. L.; Tsapatsis, M. Nat. Mater. 2006, 5, 400. (6) Hamaker, H. C. Physica 1937, 4. (7) Bergstrom, L.; Blomberg, E., Guldberg-Pedersen, H, Interparticle forces and rheological properties of ceramic suspensions. In NoVel Synthesis and Processing of Ceramics, 1999; Vol. 159-1, pp 119. (8) Bergstrom, L. AdV. Colloid Interface Sci. 1997, 70, 125. (9) Lifshitz, E. M. SoV. Phys. JETP (Engl. Transl.) 1956, 2. (10) Flatt, R. J. Cem. Concr. Res. 2004, 34, 399.
14736 J. Phys. Chem. C, Vol. 112, No. 38, 2008 (11) Cooke, D. J.; Redfern, S. E.; Parker, S. C. Phys. Chem. Miner. 2004, 31, 507. (12) Swamy, V.; Gale, J. D.; Dubrovinsky, L. S. J. Phys. Chem. Solids 2001, 62, 887. (13) Kerisit, S.; Parker, S. C. J. Am. Chem. Soc. 2004, 126, 10152. (14) de Leeuw, N. H.; Parker, S. C.; Catlow, C. R. A.; Price, G. D. Phys. Chem. Miner. 2000, 27, 332. (15) Feng, X. D.; Sayle, D. C.; Wang, Z. L.; Paras, M. S.; Santora, B.; Sutorik, A. C.; Sayle, T. X. T.; Yang, Y.; Ding, Y.; Wang, X. D.; Her, Y. S. Science 2006, 312, 1504. (16) Kerisit, S.; Cooke, D. J.; Spagnoli, D.; Parker, S. C. J. Mater. Chem. 2005, 15, 1454. (17) Sayle, T. X. T.; Parker, S. C.; Sayle, D. C. Chem. Commun. 2004, 2438. (18) Zhang, H. Z.; Gilbert, B.; Huang, F.; Banfield, J. F. Nature 2003, 424, 1025. (19) Zhang, H. Z.; Banfield, J. F. Nano Lett. 2004, 4, 713. (20) Koparde, V. N.; Cummings, P. T. J. Phys. Chem. C 2007, 111, 6920. (21) Huang, F.; Gilbert, B.; Zhang, H. H.; Banfield, J. F. Phys. ReV. Lett. 2004, 92. (22) Lewis, G. V.; Catlow, C. R. A. J. Phys. Chem.: Solid State Phys. 1985, 18, 1149.
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