Langmuir 2007, 23, 1735-1740
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Van der Waals Dispersion Forces between Dielectric Nanoclusters Hye-Young Kim,†,§,⊥ Jorge O. Sofo,†,§ Darrell Velegol,*,‡,§ Milton W. Cole,†,§ and Amand A. Lucas| Department of Physics, Department of Chemical Engineering, and the Materials Research Institute, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802, and Laboratoire de physique du solide, Facultes UniVersitaires Notre-Dame de la Paix, 61 rue de Bruxelles, B5000 Namur, Belgium ReceiVed June 22, 2006 Various methods are evaluated for their ability to calculate accurate van der Waals (VDW) dispersion forces between nanoclusters. We compare results for spheres using several methods: the simple Hamaker two-body method, the Lifshitz (DLP) theory with the Derjaguin approximation, the Langbein result for spheres, and our “coupled dipole method” (CDM). The assumptions and shortcomings of each method are discussed. The CDM accounts for all n-body forces, does not assume a continuous and homogeneous dielectric function in each material, accounts for the discreteness of atoms in the particles, can be used for particles of arbitrary shape, and can exactly include the effects of various media. At present, the CDM does not account for retardation. It is shown that even for spheres, methods other than the CDM often give errors of 20% or more for VDW dispersion forces between typical dielectric materials. A related calculation for metals reveals an error in the Hamaker two-body result of nearly a factor of 2.
Introduction Introduction to the Calculation of van der Waals Forces. Forces between colloidal particles are commonly modeled by the DLVO (Derjaguin-Landau-Verwey-Overbeek) theory.1 This theory accounts for electrostatic and van der Waals forces between particles, ignoring other forces such as depletion forces.2,3 Many studies have examined the electrostatic forces, usually according to the Poisson-Boltzmann equation.4,5 Van der Waals forces have been studied in earnest since Johannes Diderik van der Waals put forth his well-known equation of state in his PhD thesis of 1873, which won him the 1910 Nobel Prize in Physics. Recently, we have applied a method for calculating van der Waals dispersion forces (called VDW forces in the rest of this work, but always meaning dispersion forces) between nanoclusters and nanocolloids; we call this the “coupled-dipole method” (CDM).6 The purpose of this article is to assess the reliability of the assumptions in various VDW force models, especially for nanoclusters and nanocolloids, by comparing them to the CDM. VDW forces originate in the quantum-mechanical density fluctuations of electrons in the material system.7-9 At zero * Author to whom correspondence should be addressed. E-mail: velegol@ psu.edu, 814-865-8739. † Department of Physics, The Pennsylvania State University. ‡ Department of Chemical Engineering, The Pennsylvania State University. § Materials Research Institute, The Pennsylvania State University. | Facultes Universitaires Notre-Dame de la Paix. ⊥ Current address: Department of Chemistry and Physics, Southeastern Louisiana University, Hammond, LA 70402. (1) Israelachvili, J. N. Intermolecular & Surface Forces, 2nd ed.; New York: Academic Press, 1992. (2) Walz, J. Y.; Sharma, A. J. Colloid Interface Sci. 1994, 168, 485. (3) Verma, R.; Crocker, J. C.; Lubensky, T. C.; Yodh, A. G. Phys. ReV. Lett. 1998, 81, 4004. (4) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989 (with corrections 1991). (5) Sader, J. E.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 171, 46. (6) Kim, H. Y.; Sofo, J. O.; Velegol, D.; Cole, M. W.; Lucas, A. A. J. Chem. Phys. 2006, 124, 074504. (7) Margeneau, H.; Kestner, N. R. Theory of Intermolecular Forces, 2nd ed.; Pergamon: Oxford, U.K., 1971. (8) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: New York, 1976. (9) Parsegian, V. A. Van der Waals Forces; Cambridge U. P.: New York, 2005.
temperature, these are zero-point fluctuations. For interactions between clusters of atoms, there are many approaches employed in the literature for calculating the total VDW interaction (V) between bodies A and B. (1) The pairwise-sum method gives V as a summation of two-body interactions (Vij(2)) between all atoms (i) in body A and all atoms (j) in body B: NA
V)
NB
∑ ∑ i)1 j)1,j*i
NA
Vij(2)
)-
NB
∑ ∑ i)1j)1,j*i
C
,C) rij6 3p ∞ π
∫0
Ri(iω)Rj(iω)dω (1)
where rij is the separation of atoms i and j, h ) 2πp is Planck’s constant, and Rj(iω) is the polarizability of atom j at an imaginary frequency iω. NA and NB are the numbers of atoms in the clusters, and in this article we take N ) NA ) NB. The two-body summation ignores effects due to atomic screening, an inherently manybody effect. The method also ignores effects due to retardation, which is due to the finite speed of light;10 throughout this article we will ignore the effects of retardation, in part since interactions between nanoclusters are most important in the nonretarded regime. The appeal of the pairwise sum method is due to its simplicity. (2) The Hamaker method takes the continuum (integral) approach to evaluating this pairwise summation.11 The Hamaker approach has the same assumptions as the two-body summation, but it also ignores effects due to the discreteness of the atoms, and it often uses a fit coefficient (Cfit) instead of C in eq 1. Approaches 1 and 2 are often used to provide intuition into VDW forces. However, for “more precise” calculations, one usually uses (3) the Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) approach, often called “Lifshitz theory”.12 It has explicit forms for interactions between two semi-infinite bodies, two atoms, and an atom and a semi-infinite body. Results also exist for (10) Calbi, M. M.; Gatica, S. M.; Velegol, D.; Cole, M. W. Phys. ReV. A 2003, 67, 033201. (11) Hamaker, H. C. Physica 1937, IV, 1058. (12) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. AdV. Phys. 1961, 10, 165.
10.1021/la061802w CCC: $37.00 © 2007 American Chemical Society Published on Web 12/22/2006
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geometric sphere-sphere and cylinder-cylinder interactions in a vacuum.13 Lifshitz theory is applicable from the nonretarded through the retarded regimes and for all types of materials. A simple zero-temperature expression for the dispersion interaction energy per unit area between flat, semi-infinite bodies in the nonretarded regime and at separations large compared to the interatomic spacing is
V(d) ) -
A 3p ,A) 4π 12πd2
∫0∞
(
)
(iω) - 1 2 dω (iω) + 1
(2)
pi ) RiE(xi)
where d is the distance separating the bodies, and the A given here is the “Hamaker constant” evaluated by DLP theory (later called ADLP). The dielectric function (iω) for each material is often taken to be uniform throughout a material, although that will not be generally true,14 especially near to a particle surface where the atoms have a different coordination. Lifshitz theory provides results for only the few geometries mentioned previously, which are geometrically smooth. When bodies A and B are not semi-infinite bodies, but perhaps spherical clusters or colloids instead, a commonly used procedure is (4) a combination of DLP theory for semi-infinite bodies with the Derjaguin approximation (DLP-Derjaguin method).15 The Derjaguin approximation assumes that when bodies A and B are nearly touching, we can locally use the semi-infinite body DLP theory and integrate over all interacting area at the appropriate local separation. Thus, the Derjaguin method treats DLP theory as exact and then uses an additivity (inherently two-body) approach to add the contributions from various regions. The method gives the following for the interaction energy between two bodies:
VDerjaguin ) -
A 12π
∫∫S
plane
1 dxdy d(x, y)2
this, we employ our coupled-dipole method (CDM), currently valid only for the nonretarded regime, where the finiteness of the speed of light does not affect the calculations.16 The method has been outlined previously.6 While part of the method has appeared in the light scattering and other literature previously,17,18 we briefly review the CDM’s application to the calculation of VDW forces. We consider two clusters (A and B) of atoms in an applied electric field (E0), where the dipole (p) of the ith atom is given by
(3)
(4)
The electric field (E) at the location (xi) of the ith atom is not simply due to E0. Rather, the electric field consists of the applied field plus the time-varying dipole electric fields from all other atoms in both clusters, giving E(xi) ) E0 + ∑j)1,j*iNA+NB Tijpj, where NA and NB are the number of atoms in bodies A and B; the interaction tensor Tij ≡ (3nijnij - I)/rij3 for i * j and Tij ) 0 for i ) j; nij is the unit normal vector between atoms i and j; I is the 3D identity tensor; and rij is the distance between atoms i and j. This gives a set of self-consistent coupled equations for the dipoles, for each i from 1 to (NA + NB) in both clusters:
pi Ri
NA+NB
-
∑ j)1
Tijpj ) E0
(5)
We define a (3NA + 3NB) × (3NA + 3NB) matrix Q of interactions and a (3NA + 3NB) column vector (P) of dipoles such that Q3(i-1)+k,3(j-1)+m ) -Tij(km) + δkm/Ri and P3(i-1)+m ) pi(m), involving the (km) component of the ij interaction tensor (T) and the Kronecker delta function (δkm), where k or m ) 1, 2, 3 for the x, y, or z component of a vector, respectively. Now since the VDW forces exist even when E0 ) 0, we can write Q‚P ) 0. This equation has a trivial solution P ) 0, but it also has a solution when det Q ) 0. Since Q ) Q(ω), we can thus in principle find all the ω that give det Q ) 0 to find the normal modes of the system, on the basis of any model of the Ri. One simple, yet very useful, model for the atomic polarizability is the Drude model,19
where Splane is the plane perpendicular to the line connecting the two particles, and x and y are positions in this plane. The local Hamaker constant (A) depends in general upon the distance between the bodies (because of retardation), but since in this article we ignore the effects of retardation, we put A outside the integral. The DLP-Derjaguin approach allows for media between two spherical or cylindrical particles. Exact results do not exist for sphere-sphere or cylinder-cylinder interactions in a medium, and it would require additivity approximations similar to those used in the latter part of Hamaker’s paper.11 Although it is often assumed that Lifshitz calculations for VDW forces are accurate for all cluster or particle sizes at all separations, in fact two key limitations are that interaction formulas exist for only a few shapes of particles, and discrete atom effects are ignored. Both of these are important considerations for VDW forces between nanoclusters, which often have nonspherical and atomically rough shapes. In addition, the exact results (e.g., the sphere-sphere result) are usually not used; rather, approximations like the DLP-Derjaguin method are often used. This approximation ignores changes of the dielectric function near to the surface, as well as discrete-atom effects. The Coupled Dipole Method. In this article, we assess the extent to which the assumptions in these previous methods limit their accuracy for nanocluster or nanocolloid systems. To do
where R0i is the static polarizability and ω0i is the characteristic frequency of atom i. One can view eq 6 as a two-parameter fit to R(iω) data, although a physical basis exists for its form. As discussed earlier in this paper, we could use a more exact form for the polarizability, but one often finds that a single ultraviolet contribution to the VDW forces dominates other resonant frequency contributions for an atom. Setting E0 ) 0 and substituting the Drude model into eq 4 gives ω0i2pi - ω2pi ) R0iω0i2 ∑j)1,j*iNA+NB Tijpj, and by defining Ω3(i-1)+k,3(j-1)+m ) - R0iω0i2Tij(km) + ω0i2δijδkm for the km component of the ij interaction, we obtain Ω‚P ) ω2P. This eigenvalue problem is readily solvable using standard algorithms, assuming that the number of particles is sufficiently small that the relevant matrices can be diagonalized; in the present study, the matrices are of dimension 14 040 × 14 040 (since N ) 2340 particles). The
(13) Langbein, D. Theory of Van der Waals Attraction; Springer Tracts in Modern Physics, Volume 72; Springer-Verlag: New York, 1974. (14) Kiefer, J. E.; Parsegian, V. A.; Weiss, G. H. J. Colloid Interface Sci. 1975, 51, 543. (15) Hunter, R. J. The Theory of van der Waals forces. In Foundations of Colloids Science, Vol. I.; Clarendon Press (Oxford University Press): New York, 1986 (with corrections in 1992); Chapter 4.
(16) Casimir, H. B. G.; Polder, D. Phys. ReV. 1948, 73, 360. (17) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley: New York, 1983. (18) Yang, W.-H.; Schatz, G. C.; Van Duyne, R. P. J. Chem. Phys. 1995, 103, 869. (19) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Brooks/Cole: Belmont, CA, 1976.
Ri(iω) )
R0iω0i2 ω0i2 + ω2
(6)
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eigenvalues give the square of the mode frequencies (ω2). Since each frequency contributes pω/2 of energy (or at finite temperature, pω/2 coth(pω/2kT)), we can find the energies for the isolated clusters A and B and also for the combined system of cluster A near cluster B. Taking the difference of the total system minus the two independent clusters gives the “exact” interaction energy, subject only to the assumptions that the Drude model describes the atomic polarizabilities, that only dipole interactions are important, and that retardation is not important. A previous paper6 demonstrates how the CDM gives not just two-body and three-body forces but all n-body forces computed explicitly, an unusual situation in condensed-matter physics. In our calculations, we consider only dipole-dipole interactions (i.e., dipole polarizabilities), neglecting higher-order multipoles. While the higher-order multipoles might be expected to play an important role for neighboring atoms, and therefore, intracluster interactions or cohesion problems, quadrupole-dipole and other higher-order interactions would play a small role when the intercluster atom-atom distances are large compared with the size of the atoms (i.e., the VDW energy problem). Furthermore, since the VDW energy is the difference in the calculated energies of the total system and of the two isolated clusters, any errors in the intracluster energies are subtracted out. We have previously used the Axilrod-Teller-Muto (ATM) approach to calculate the first correction to the two-body sum of eq 1.20-22 However, the ATM approach neglects four-body corrections and higher, which we now often find to be important.6 In fact, as shown here, the addition of only a three-body force is often misleading, giving a result that has a greater error than the simple two-body summation. Defining a Comparison of Methods. Having listed the alternative methods of determining the VDW energy, we must define how to compare them. To do this, we must first be able to relate the bulk permittivity (iω) to the more fundamental, atomistic parameter nR(iω), which is the product of the number density of the system and the polarizability at imaginary frequencies. For dielectric systems, we expect nR(iω) to remain constant, regardless of cluster size, although the CDM can readily incorporate a spatially varying nR(iω). We use the ClausiusMossotti (CM) relation 4πnR(iω)/3 ) [((iω) - 1)/((iω) + 2)] for each material to relate the molecular and bulk parameters.19 The CM relation is a quasi-static, continuum equation. The quasistatic assumption is valid here because of the small size of the nanocluster, even at ultraviolet frequencies.17 We will later use the polarizability obtained from the CM relationship to show that the bulk permittivity is not valid up to the surface. This procedure is valid since the CM relationship gives the molecular polarizability in the bulk. This molecular polarizability remains nearly constant down to the nanocluster size, which enables us to show that the bulk permittivity does not. We approximate the polarizability with the Drude model in eq 6. More complicated expressions could be used, with only small changes in our overall results. Using the Drude model in the CM relationship, one finds that
(iω) ) 1 +
4πnR0/(1 - 4πnR0/3) 1 + (ω/ω0)2/(1 - 4πnR0/3)
(7)
Two important points from this expression are that (1) a well-
(20) Axilrod, B. M.; Teller, E. J. Chem. Phys. 1943, 11, 299. (21) Muto, Y. Proc. Phys.-Math Soc. Japan 1943, 17, 629. (22) Gatica, S. M.; Cole, M. W.; Velegol, D. Nano Lett. 2005, 5, 169.
Figure 1. Permittivity ratio [(iω) - 1]/[(0) - 1] and polarizability ratio R(ω)/R0 as functions of the reduced frequency ω/ω0. Many nonmetallic materials have nR0 < 0.10 for the ultraviolet resonance peak, which contributes most to the VDW forces. Typically, the characteristic frequency ω0 is derived from the ionization energy.
known “polarization catastrophe” occurs at nR0 ) 3/4π ) 0.239,23 but in this article we use much less electronically dense systems; and (2) the resonant frequency for the polarizability is different from that for the permittivity by a factor x1-4πnR0/3 . Our nR0 ) 0.0639 (n ) 0.022 A-3 and R0 ) 2.907 A3) and ω0 ) 2.376 × 1016 rad/s, giving a resonant frequency for the permittivity of 2.033 × 1016 rad/s. For our parameters, the expression above gives (0) ) 2.096. The raw data available for several systems is the bulk permittivity as a function of frequency (or imaginary frequency). Our approach neglects the infrared (IR) and other peak frequencies in the permittivity, which usually give small contributions to the VDW forces, but we could include these if they were important. Thus, for fused silica the actual (0) ) 3.81, but the neglect of the IR contribution would give 2.096. When the IR contribution of the permittivity is neglected, since it contributes little to the VDW forces for most materials, typical nR0 ) 0.05-0.10, except for metallic materials. The permittivity and polarizability ratios are given in Figure 1 for various nR0. Having these relations between the molecular and bulk properties, we now present two approaches for calculating a Hamaker constant. (a) We convert nR(iω) to the bulk parameter (iω) and then use DLP theory (eq 2) to calculate a Hamaker constant (ADLP):
ADLP )
3π2pω0(nR0)2 4(1 + 2πnR0/3)3/2
(8)
A second method is (b) to use nR(iω) in eq 1 to find C ) 3pω0R02/ 4. Hamaker showed (eq 13a of ref 11) that the constant in his name is given by (nπ)2C, and so the two-body summation value for the Hamaker constant is
Asum )
3π2pω0(nR0)2 4
(9)
The expression for Asum can equivalently be found by expanding ADLP in nR0, since in the limit nR0 ) 0 many-body interactions disappear. Asum can also be found by directly integrating the pair (23) Kittel, C. Introduction to Solid State Physics, 8th ed.; Wiley: New York, 2004.
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Figure 2. Ratio of various methods for computing VDW forces to the exact CDM method, for two identical spheres with N ) 2340. The center-to-center distance between the spheres (d) is the same for both the continuum and discrete methods. The cluster diameters are both 5.88 nm. Because of the discrete placement of the atoms within the cluster, analytical methods (e.g., Langbein, Hamaker) here far overestimate the result for small gaps, although the Langbein result is within 2% by the time the gap is 3 nm. The two-body sum with C approaches the exact CDM value at far distances, since multibody effects disappear. At much larger separations than shown on the figure, the CDM and eq 11 converge to the same result, as expected. At far distances, the use of Cfit predicts a VDW interaction that is smaller than the exact value. Results from the Derjaguin approximation are far off this figure for the parameters listed; typically, it overestimates V by a factor of 5 or more.
potential of eq 1 over the two-half spaces. Thus,
ADLP 1 ) Asum (1 + 2πnR /3)3/2 0
(10)
For our two-body direct summation (eq 1), we will use C defined above, and we will also use a fit value Cfit ) ADLP/(nπ)2. The latter is another commonly used approach in the literature, and the assumption is that since DLP accounts for many-body effects, fitting for Cfit from ADLP will give the accuracy of DLP with the simplicity of the two-body sum. Our calculations will show that this procedure is not correct. The equivalent error is to use ADLP in the Hamaker integral result. Effectively, this procedure “over accounts” for the many-body effects, since ADLP comes from the interaction of two semi-infinite bodies. We note that C/Cfit ) Asum/ ADLP > 1.
Results and Discussion Figure 2 shows the results for VDW interactions (V) between two identical spheres with N ) 2340 (diameter 5.88 nm), in a face-centered cubic (FCC) lattice with a nearest neighbor distance of 0.4 nm and R0 ) 2.907 A3. Smaller values of N were also studied, with similar qualitative results. The Hamaker integration approach using ADLP asymptotes to a value 17% below the exact; the use of Asum would cause the Hamaker approach to asymptote to nearly the exact value. The sphere-sphere result (listed as “Langbein sphere solution”) asymptotes properly at large distances but fails at smaller distances because of the discrete placement of atoms. That is, in both cases the continuum approximation fails at close distances, and such effects would be worse with surface roughness or defects. On the other hand, the discrete summations remain somewhat close (within 20%) to the exact CDM result. The discrete summation using C asymptotes properly at large separations (since the integral and sum then coincide), while the discrete summation with Cfit does not. This distinction is important, because it is known that manybody terms become less important at larger distances for the
spherical geometry.24 Essentially, the use of DLP theory to obtain a “fit C” is not correct, because the method is derived from the many-body screening for semi-infinite slabs, which overestimates the many-body screening for the case of spheres. The results of these two methods differ by about 21%, as expected from eq 10. The use of DLP-Derjaguin gives values of V up to a factor of 5 too high, and these values do not even appear on the figure. The primary failure is due to the Derjaguin and continuum approximations. For two spheres, the discrete two-body sum is close to the exact result; the difference between them is always less than 15% over the complete range of separation. This implies that many-body effects are small. Why is that the case? At very large separation, r . R, the spheres may be treated as point particles. In that case, their interaction can be evaluated from the London formula applied to the sphere as a whole. In previous work, we found that many-body effects disappear for large spheres,24 so that the sphere’s polarizability is NR(iω). In that case, the spheresphere interaction is given exactly by the two-body sum. Asymptotically, this takes the form
limV(r) ) -CN2/r6
rf∞
(11)
For the system in this article, the value C ) 1.588 × 10-77 J-m6 and N ) 2340 for each cluster. The curve representing this asymptotic expression is plotted in Figure 2 in relation to the exact V; only at very large r do the two results converge. As the spheres get close, interacting subvolumes of closest proximity cause the interaction to greatly exceed the form given by this asymptotic relation, which concentrates the response at the center of the sphere. We may analyze the behavior at close separation as follows. For the regime (r - 2R) , R, the interaction is dominated by neighboring regions of near contact of the spheres and even neighboring atoms. Continuum approaches fail at this close approach, and the impact of retardation is small since most of the interaction arises near the contact region. At these close separations, when the interaction is somewhat well-approximated by two flat, semi-infinite half spaces, the discrete sum approach using Cfit yields a better approximation to the exact potential than that using C (Figure 2). The alternative approximations to these sums yield unsatisfactory results in most cases. Both versions of the Derjaguin model fail badly, vastly overestimating the interaction, and are out of the range of Figure 2. This is not really surprising because this model assumes that the interacting bodies are semi-infinite. That is a particularly poor approximation at large separations (d). The integrated Hamaker expression works well at large d. However, it fails badly at small d because of the omission of the discrete lattice effects. Figures 3 and 4 show the results for two cylinders, oriented either parallel or crossed, with N ) 2340 and FCC packing. The essential conclusions are the same as in Figure 2. No analytical “Hamaker integration” scheme exists for cylinders, although an exact result exists for continuum, geometric cylinders.13 However, the use of the exact result for geometric cylinders is very difficult to define for this case, since the discreteness of the atoms effect becomes even more pronounced than in Figure 2 for spheres. This is because the diameter of the spheres is less than 10 atoms, and defining a proper diameter for the Langbein cylinder result simply becomes a fitting game. As for the Derjaguin approxima(24) Gatica, S. M.; Calbi, M. M.; Cole, M. W.; Velegol, D. Phys. ReV. B 2003, 68, 205409.
Van der Waals Dispersion Forces
Figure 3. Ratio of various methods for computing VDW forces to the exact CDM method, for two identical, parallel cylinders with N ) 2340. The diameter of the cylinders is 2.30 nm, while the length is 25.6 nm. The calculations give similar conclusions to Figure 3. The scale is expanded compared to Figure 2.
Langmuir, Vol. 23, No. 4, 2007 1739
CDM can readily incorporate a medium other than vacuum between A and B. Interaction between Metallic Spheres. In the preceding discussion, we explored interactions between dielectric bodies. For interacting spheres, we found that the Hamaker approximation underestimates the asymptotic interaction by about 20%. The reason is that the effective interaction coefficient in that approach (Cfit) is derived from the exact interaction between two slabs, with parallel surfaces. This coefficient is smaller than the “bare” coefficient C because of dielectric screening present in the slab geometry. When Cfit is used to compute the interaction between two spheres, however, the screened value is not appropriate because screening is absent from the intersphere problem at large separations. The preceding argument concerning the inadequacy of the Hamaker approach is quite general. Indeed, one can further assess its reliability by undertaking a similar test for the extreme case of two interacting free-electron metal spheres. For specificity, we focus on the regime of separation (d) much greater than their radii R (but not so large that retardation occurs). In that case, we may define an interaction coefficient (K) from the known powerlaw form of the asymptotic interaction, as
K ≡ -r6V(r)
(12)
Schmeits and Lucas applied a continuum coupled dipole method similar to that in this article to show26
K ) (31/2/4)pωpR6
Figure 4. Ratio of various methods for computing VDW forces to the exact CDM method, for two identical, crossed cylinders, separated by an axis-to-axis separation dc, with N ) 2340. The diameter of the cylinders is 2.30 nm, while the length is 25.6 nm. The calculations give similar conclusions to Figure 2.
tion, similar to the case of spheres, it gives overestimated results which are far above the range of Figures 3 and 4. A commonly employed procedure is to use the DLP value of A in the Hamaker calculation for two spherical clusters. This procedure perhaps works well for larger particles (e.g., 100s of nm in diameter), but for small clusters, the assumption that two semi-infinite bodies are interacting locally fails. In addition, the screening effects present in the calculation for semi-infinite bodies are less prevalent in spherical clusters. Thus, the use of the DLP value for A, which includes all screening effects, gives results that are roughly 20% in error for our choice of R0 and ω0. Another important point concerns interactions at close separations. Using the DLP value for A assumes that the permittivity is correct up to the surface of the bodies. However, we expect that the molecular property (R) remains more constant than the bulk property (), giving an error when using the DLP approach. This is especially the case for nanoclusters, where a much greater fraction of the atoms are close to the surface. We have found previously that atoms within roughly 10 atomic diameters of the surface behave differently from the bulk.25 Another virtue of the CDM is that one can incorporate more subtle effects, such as the rearrangement of atoms at surfaces or edges or corners, because of strong intracluster cohesion forces. One simply alters the locations of the atoms. Such rearrangements are difficult or not possible with other methods. Finally, although we have not done it here, the (25) Kim, H. Y.; Sofo, J. O.; Velegol, D.; Cole, M. W.; Mukhopadhyay, G. Phys. ReV. A 2005, 72, 053201.
(13)
where pωp is the (bulk) plasmon energy of the metal. This equation gives the nonretarded dipole-dipole VDW coupling of two distant metallic spheres, each represented as free-electron continuum having a bulk plasma frequency (ωp) and therefore a triply degenerate dipole plasmon of frequency ωp/x3 and polarizability R3. There is no further approximation. For comparison, we evaluate the corresponding Hamaker value of the interaction coefficient, KHam. To do so, we first determine the Hamaker coefficient from the known interaction energy, per unit area, of two semi-infinite slabs separated by a distance d, using the integral (eq 2) for the coefficient A and the dielectric function of a freeelectron metal:
(iω) ) 1 + (ωp/ω)2
(14)
We obtain A ) 3pωp/29/2 and Cfit ) 3pωp/29/2(nπ)2. At large separation between spheres, the intersphere interaction coefficient (within the Hamaker approximation) is expressed in terms of the number of (assumed monovalent) atoms in each sphere, N ) 4πR3 n/3.
KHamaker ) N2Cfit )
pωpR6 3x2
(15)
This result may be compared with the exact value in eq 13. Their ratio is
K KHamaker
)
(23)
3/2
) 1.84
(16)
Hence, the Hamaker interaction between metal spheres is nearly a factor of 2 smaller than the exact interaction. Recall that the corresponding ratio is 1.21 for the dielectric studied above. The (26) Schmeits, M.; Lucas, A. A. Prog. Surf. Sci. 1981, 14, 1.
1740 Langmuir, Vol. 23, No. 4, 2007
origin in both cases is that the Hamaker coefficient includes screening, on the basis of the slab-slab problem, which is absent from the sphere-sphere problem. The magnitude of the discrepancy is larger for a metal than a dielectric because of the larger polarizability.
Conclusions In this article, we have assessed a variety of approximate methods of computing VDW interactions for various geometries, comparing them with exact results from the CDM. For dielectric materials with nR0 ∼ 0.06, the discrete sum of two-body interactions works relatively well in computing the interaction energy for several geometries, including spheres and cylinders separated by a vector perpendicular to their orientation. In other cases (cylinders separated by a vector parallel to their orientation and free-electron metals), the Hamaker approach, or two-body summation, is a poor method, with errors of order 100% for many situations. The distinction between favorable and unfavorable geometries for interacting cylinders occurs because manybody effects greatly increase the polarizability of a rod when the field is oriented parallel to it. When two cylinders are end-to-end collinear, the VDW attraction is greatly enhanced because of the many-body effect. This failure of the two-body sum is found for all separations, as was shown in a previous work.6 While we have examined spheres and cylinders in vacuum in this work, the CDM is applicable to any particle shape and can have any medium between the particles. The other case of large many-body effects involves freeelectron metals. The key point for such a situation is that the dominant resonance in the dynamical response of the sphere occurs at ωp/31/2. In contrast, the interaction between two planar
Kim et al.
half-spaces is mediated by surface plasmons, with a resonant frequency ωp/21/2. In the Hamaker approach, the value of the nominal pair interaction coefficient Cfit is deduced from the latter geometry, giving rise to a large discrepancy (3/2)3/2 ) 1.84 in the predicted sphere-sphere interaction. The DLP-Derjaguin method, often used by researchers for calculating VDW forces between spherical colloids, has been found to be quite inaccurate for nearly all situations explored here with nanocolloids. The assumptions of a uniform dielectric function (even up to the surface), the absence of discrete atom effects, and interacting semi-infinite bodies all fail for the nanocolloid case. Continuum Lifshitz theory results for spheresphere interactions often fail because of the discrete placement of the atoms. Currently, experimental evidence and methods are lacking to test calculations for VDW forces. In the absence of experimental data, the CDM provides a viable technique to predict VDW forces and to gauge other methods for accuracy. If one prefers to use a simpler model than the CDM, one might often choose to use the two-body discrete sum, but with C instead of Cfit. As a byproduct of the present CDM calculations, we have determined collective modes (plasmons) localized on the surface or edge of the various clusters. In future work, we will address the electron energy loss spectrum (EELS) excited in the dielectric clusters by an incident electron. Acknowledgment. This research has been supported by the National Science Foundation (grants DMR-0505160 and NER CTS-0403646). LA061802W
