NANO LETTERS
Designing van der Waals Forces between Nanocolloids
2005 Vol. 5, No. 1 169-173
Silvina M. Gatica,† Milton W. Cole,†,§ and Darrell Velegol*,‡,§ Department of Physics, Department of Chemical Engineering, and Materials Research Institute, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16803 Received October 21, 2004; Revised Manuscript Received November 24, 2004
ABSTRACT van der Waals (VDW) dispersion forces are often calculated between colloidal particles by combining the Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) theory with the Derjaguin approximation; however, several limitations prevent using this method for nanocolloids. Here we use the Axilrod-Teller-Muto 3-body formulation to predict VDW forces between spherical, cubic, and core−shell nanoparticles in a vacuum. Results suggest heuristics for “designing” nanocolloids to have improved stability.
Introduction. van der Waals (VDW) dispersion forces between colloidal particles have been calculated using Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) theory1,2 for over 30 years. The usual scheme is to combine DLP3,4 with the Derjaguin approximation5,6 to account for particle curvature with spherical or rod-shaped particles. For nanoparticles, this method of calculation has several critical shortcomings. (1) Accurate limiting cases can be difficult to evaluate except for particles either nearly-touching or far apart.7-9 For intermediate separations, a common approximation is to use an additive Hamaker approach.10 (2) The dielectric or polarizability properties for nanocolloids are neither bulk nor molecular,11-13 and even within a particle can be spatially varying. (3) The discrete nature is usually ignored for the constituent atoms in the nanocolloids or nanocluster. (4) DLP provides little mechanistic insight into how to design more stable nanocolloids.14 In this letter we use the Axilrod-Teller-Muto (ATM) 3-body formulation15-17 to predict VDW forces between spherical, cubic, and core-shell nanoparticles18 in a vacuum (Figure 1). We focus on points 1, 3, and 4 from the Introduction. Our previous research has addressed point 2,19 and we expect this to be an important avenue of future research. The long-term goal of the work is to develop heuristics for “designing” nanocolloids to have the desired dispersion and assembly (e.g., quantum dots20 and fluorescent particles) by examining all four points. Method for Evaluating VDW Forces. A general formalism for calculating VDW forces is to consider atom-atom * Corresponding author. E-mail
[email protected]. † Department of Physics. ‡ Department of Chemical Engineering. § Materials Research Institute. 10.1021/nl048265p CCC: $30.25 Published on Web 12/17/2004
© 2005 American Chemical Society
interactions (two-body interactions), then three-body interactions, four-body interactions, etc. This is written3,4 V0 )
(3) (3) + ∑∑Vijγ ) + ... ∑i ∑γ Viγ(2) + (∑i β>γ ∑ Viβγ i99.9% Al2O3), and water.8 The “atomic” polarizabilities are taken for the entire unit (e.g., SiO2, Al2O3). The dielectric spectra come from absorption measurements, giving the loss modulus (′′) at real frequencies (ω); a Kramers-Kronig relation then transforms this function to the real function (iω). The complexity of the spectra (particularly, water) makes it difficult to use a simple Drude model. In this manuscript we have neglected changes and spatial variations in polarizabilities (R) due to the nanosize of the particle, and we make a further estimate, obtaining the polarizability from (iω) using the Clausius-Mosotti relationship22 (iω) - 0 (iω) + 20
)
4π n R(iω) 3 0
(4)
While this equation makes no particular assumption about the form of the dielectric function (i.e., it does not depend on a Drude model), nor does the model depend on the substance density (n0), the equation does assume that the material is a continuum. We recognize that the continuum approximation is not correct for nanocolloids, since there are so many surface atoms compared with interior atoms, but combining this approximation with known spectra is the best approximation available for the polarizabilities of these nanoclusters. Table 1 lists values of the polarizabilities and number densities for the atoms used in this letter. The equation used to construct the function (iω) is23
(iω) ) 1 +
dj
∑j 1 + ωτ + ∑j j
170
fj ω
() ω
1 + gj + ωj ωj
2
(5)
Figure 2. VDW forces between two silica spheres with n ) 619 atoms in a vacuum. The lattice constant a ) 3.569 Angstroms and the distance of closest approach occurs for a center-to-center distance of 10.192a. While the two-body forces capture the qualitative trend of the VDW forces, the three-body forces are essential for quantitative results, since they constitute roughly 20% of the total VDW forces.
where the molecular dipoles (dj), the relaxation times (τj), oscillator strengths (fj), resonance frequencies (ωj), and bandwidths (gj) are known24 for the substances we examined. Table 2 lists our calculated values of the I6 and I9 coefficients defined in eq 3. The adequacy of using only two-body and three-body terms in eq 1 depends on having particles for which the atomic density is sufficiently small. The authors have previously shown that the two-body and three-body terms are the first two terms in an expansion of DLP theory,17 and this has been verified by others.14 The two-body and threebody interactions ignore quadrupole, octupole, and higher order terms, and thus anisotropies that will arise at short distances compared with the atomic radius. In addition, at very small distances, four-body and higher atom interactions can become important, along with higher order terms in the separation (e.g., r-8 and r-10). It must be remembered, however, that cluster separations smaller than the cluster size can still be large on the atomic scale. As the figures in this letter show below, the two-body and three-body VDW forces capture the essential physics of many real material systems. Results and Discussion. Figure 2 shows for silica spheres the two-body, three-body, and two-body-plus-three-body VDW interactions. An important point is that at intermediate gaps, the three-body forces are as much as 21% of the twobody forces. Thus, it is important to account for the threebody forces for quantitative purposes. Furthermore, this ratio (∼0.2) suggests that the four-body forces are likely to be Nano Lett., Vol. 5, No. 1, 2005
Table 2. Evaluated Constants I6 and I9 for Equation 3a
a
system
I6 (A6/s)
system
I9 (A9/s)
silica-silica silica-water silica-hexane silica-sapphire sapphire-sapphire sapphire-water sapphire-hexane water-water water-hexane hexane-hexane
1.635 × 1017 0.816 6.036 2.456 3.698 1.228 9.043 0.409 3.005 22.44
silica-silica-silica silica-silica-water silica-silica-hexane silica-silica-sapphire silica-water-water silica-water-hexane silica-water-sapphire silica-hexane-hexane silica-hexane-sapphire silica-sapphire-sapphire water-water-water water-water-hexane water-water-sapphire water-hexane-hexane water-hexane-sapphire water-sapphire-sapphire hexane-hexane-hexane hexane-hexane-sapphire hexane-sapphire-sapphire sapphire-sapphire-sapphire
3.747 × 1017 1.863 13.87 5.564 0.931 6.870 2.769 51.85 20.54 8.275 0.469 3.415 1.383 25.62 10.18 4.12 195.3 76.63 30.46 12.32
The Clausius-Mosotti equation was used to estimate molecular polarizabilities from known dielectric data and expressions.24
Figure 3. VDW potentials between two cubic particles with the given center-to-center separation (r/a). The cubes have 125 silica atoms. The inset shows two of the many possible orientations (top inset shows face-to-face; bottom inset shows corner-to-corner). In the limit of large r/a, eqs 1-3 and I6 from Table 2 can be used to find that the asymptotic ordinate value is -777 eV, since for large r/a the three-body interactions should approach zero.
much smaller (i.e., ∼0.22). At very large gaps, we have shown previously (numerically and analytically) that threebody forces for spherical particles become insignificant compared with two-body forces, for reasons of symmetry.19 We emphasize that the current accuracy limitation in Figure 2 and in other calculations in this paper results primarily from the accuracy of the available polarizability data, not from neglecting four-body and higher forces. Our research group continues to study changes in polarizability (including spatial variations within the particles) for nanocolloids compared with bulk or molecular values. Figures 3 and 4 compare forces between silica spheres and cubes. Figure 3 shows the VDW potential for two cubes averaged over all orientations, while Figure 4 compares potentials for cubes at various orientations, and also for Nano Lett., Vol. 5, No. 1, 2005
Figure 4. Interaction energies between particles having various relative orientations. All cases are normalized by the energy of two cubes averaged over all orientations. The cubic particles have 125 silica atoms, while the spheres have N ) 123.
spheres. For purposes of the comparison, the cube has N ) 125 atoms, and the sphere has N ) 123 atoms (i.e., nearly the same). The small number of atoms gives small VDW attractions (∼kT/15 at small separations), but enables calculation of VDW forces over many orientations in a reasonable computation time. Nevertheless, it is important in viewing Figure 3 to remember that the VDW interaction scales roughly as N2. The eventual goal is to test heuristics learned from these calculations experimentally, and thus going from N ) 125 atoms to N ) 2500 atoms would give attractive VDW interactions of roughly 25kT for the system shown. The interparticle distance (r) is center-to-center, since it would otherwise not be possible to define the gap for the various orientations of cubes. The cubes are examined for several geometries: (1) when the faces are parallel to each other, (2) when the corners of the cubes give the point of closest approach, (3) when the edges of the cubes give the 171
Figure 5. VDW forces with various shells on a silica core, relative to values for a particle of the same radius made from pure silica. The core has N ) 515 atoms, while the shell has N ) 104 atoms. Three-body forces are roughly 30% of the total energy. A layer of hexane causes the spheres to attract more than a layer of water.
closest approach, and (4) averaged over all orientations of particles 1 and 2, such that 〈V〉 )
∫ ∫∫ ∫ 2π
0
π
0
2π
0
π
0
V(θ1, φ1, θ2, φ2)sin θ1 sin θ2 dθ1 dφ1 dθ2 dφ2 (4π)2
(6) For a view of two of the orientations, see the inset to Figure 3. The orientations of the cubes were specified by identifying the coordinates of every constituent of the cube, and then rotating the coordinates about two independent axes using tensor rotation matrices.25 The face-to-face orientation of the cubes gives the smallest VDW attraction for a fixed centerto-center distance; the corner-to-corner orientation gives the largest attraction (as expected, because this combination has the closest approach). Thus, if the angular orientation of the particle is fixed, a cubic shape gives either the largest or the smallest attraction, depending on the orientation. On the other hand, if Brownian motion is able to randomly orient the cubes, then the cube has more attraction than the sphere. Figure 5 compares interactions between spherical silica particles with various shell layers. To simplify the calculation, we made all atoms have the same size and scaled the real atomic polarizability of the shell material to the polarizability used in the calculation. The relation is Rcalc shell ) /n , where the n is atomic density. As expected, nshellRreal core shell the core particles with the lower polarizability material in the shell layer have smaller VDW attractions. Adding the three-body contribution is important for these systems, since it is up to about 30% of the two-body value. Importantly, the water shell gives smaller VDW attractions than the hexane shell. While calculations for Figure 5 were done in a vacuum, the calculations also have ramifications for particles in a liquid environment. Clearly, if the particles are in vacuum, those with shells have greater mutual attraction than particles without a shell. However, if the nanocolloids are immersed in a solvent, then either that solvent surrounds the particle or some other shell of material surrounds the particle. By 172
making the shell material with lower polarizability than the bulk solvent that otherwise would surround the particle, we can reduce VDW attractions between the particles, as shown in Figure 5. The results indicate that a cosolvent system, with a very small amount of a second solvent dissolved in the primary solvent, might greatly improve nanocolloid stability in three ways. (1) The majority solvent can be chosen to minimize VDW attractions. For example, putting silica particles in pure octane instead of pure water can reduce VDW attractions by a factor of 5.26 (2) If the minority cosolvent selectively binds to the surface (e.g., water, for silica particles in an octane-water mixture), the adsorbed layer will reduce the VDW attractions further, as Figure 5 shows. (3) The adsorbed minority cosolvent can add a repulsive solvation force.27,28 All three effects lead to improved particle stability. Yet a fourth possibility is that the adsorbed cosolvent will reduce surface reactions or dissolution, minimizing Ostwald ripening of the particles.29-31 Currently, we are working on measurements of nanocolloid forces,32 in order to test our hypothesis concerning cosolvents. The ATM method is relatively simple to use for many material systems. Computational constraints can, of course, make the calculations time-consuming, since the number of terms in the ATM model grows as N3 rather than as N2 (as for two-body systems). In addition, the ATM method is quite amenable to using calculations of more exact polarizability data33,34 or calculations35,36 for nanoclusters. For denser systems, four-body forces and others will need to be considered; however, the three-body forces often give reasonably quantitative results that can be used to design particle systems. It is important to note that direct numerical calculations using density functional theory37 or other methods usually give energies with insufficient accuracy to determine VDW energies; however, these methods can give polarizabilities with sufficient accuracy to use with the ATM method.13,19,36 Our calculations have not been compared with DLP theory, since neither the nearly-touching limit (i.e., the Derjaguin approximation) nor the far-field limit (i.e., r-6) apply. That is, even at small gaps, where the Derjaguin approximation would normally work, for nanocolloids the distance required for the approximation is less than the distance between atoms, invalidating the DLP model. However, previous investigators have examined VDW interactions between two spheres or clusters at arbitrary separations. Langbein developed a general expression for the nonretarded VDW interactions between two spheres at any separation,38 and Kiefer et al.39 simplified Langbein’s expression to a more easily calculated form. However, these works rely on having geometrically perfect spheres, rather than clusters or particles with discrete atoms. These authors have also examined the case of particles having internally varying polarizabilities (and thus, continuum core-shell particles).8,40 Marlow and co-workers have carried out extensive calculations of cluster VDW interactions, including for “discrete” systems and complex shapes.41,42 They have compared results of alternative methods: (a) a continuum approach Nano Lett., Vol. 5, No. 1, 2005
based on Lifshitz theory, (b) Hamaker theory for the spatial variation, with a substitution of the asymptotic many-body coefficient for the two-body interaction coefficient, and (c) the VDW interaction derived from a set of coupled harmonic oscillators. Each of these approaches has its merits and disadvantages, and a definitive comparison is not feasible here. Method (a) is suitable only at separations large compared to the lattice spacing, where DLP theory applies. Method (b), which is widely used, is known to exhibit qualitative errors in related applications.43 Method (c) includes all orders of particle interactions, but it is limited to systems that follow the harmonic oscillator model (i.e., none of the systems studied in this letter) and is computationally more expensive than our approach. Marlow and coworkers have also evaluated a fourth model, a sum over molecules of particle A of the individual many-body interactions with particle B. While appealing in some respects (e.g., accuracy at small separation), this approach does not yield the known long-range interaction. Conclusions. Two important, distinct, and yet equivalent approaches in the literature for calculating VDW interactions are (1) evaluating the effect of the perturbation of the free electromagnetic field caused by the presence of the particles (i.e., the Lifshitz approach) and (2) adding the two-body (Hamaker), three-body (Axilrod-Teller-Muto), four-body, etc. interactions. We have used the second approach, since it is computationally tractable for nanocolloids and since it provides mechanistic insight. The ATM method provides a powerful method for using polarizabilites to calculate the leading many-body contributions to VDW forces. Furthermore, the method overcomes some of the limitations of current evaluations of the Lifshitz theory. In sum, the calculations in this letter enable us to hypothesize two heuristics that could lead to improved particle stability: (a) use spherical particles instead of cubic if the cubic particles can assume all orientations, and (b) use a cosolvent system where one solvent selectively adsorbs to the particle, creating a “shell” of low polarizability (small attraction), and also a possible solvation layer (large repulsion). Acknowledgment. The authors thank the National Science Foundation (NER grant CTS-0403646, NIRT grant CCR-0303976) and the Ben Franklin Center of Excellence for funding this work. We thank Jorge Sofo for helpful discussions. References (1) Lifshitz, E. M. SoV. Phys. JETP 1956, 2, 73. (2) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. AdV. Phys. 1961, 10, 165. (3) Margenau, H.; Kestner, N. R. Theory of Intermolecular Forces, 2nd ed.; Pergamon: New York, 1971.
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