ARTICLE pubs.acs.org/JPCC
Spatial Nonlocality in the Calculation of Hamaker Coefficients R. Esquivel-Sirvent*,† and George C. Schatz‡ † ‡
Instituto de Física, Universidad Nacional Autonoma de Mexico, Apartado Postal 20-364, D.F. 01000, Mexico Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States ABSTRACT: We study the effects of spatial dispersion or nonlocality in the calculation of the Hamaker coefficient. Using a hydrodynamic model to describe the dielectric function of the materials, we show that at small separations, less that 10 nm, the Hamaker coefficient decreases significantly when a nonlocal dielectric is used. As a case study we compute the van der Waals interaction between two Au nanospheres and between two Au parallel nanorods. In both cases, the interaction energy shows variations of up to 2 orders of magnitude between the local and nonlocal cases. We show that at the nanoscale the usual approach to the calculation of the Hamaker coefficients has to be modified to properly take into account the dielectric properties of the nanoparticles.
’ INTRODUCTION Self-assembly processes, colloidal stability, and wetting phenomena are all related to the existence of van der Waals forces. For colloidal particles and nanostructures where the original molecular van der Waals force is no longer applicable, Hamaker developed a pairwise approximation to deal with extended bodies. This approach was later improved by the Lifshitz continuum theory,1,2 considering that the dispersive force comes from temperature fluctuations and zero-point fluctuations of the electromagnetic field. The Lifshitz expression for the force between two parallel slabs can be obtained from the fluctuationdissipation theorem, but the optical properties of the materials over a wide frequency range must be known. This means that the frequency-dependent dielectric function ε(ω) must be available. For simplicity, in the rest of the paper we assume nonmagnetic materials and assume a magnetic susceptibility μ = 1. In the Liftshitz theory calculation of van der Waals interaction between two slabs separated by a distance L, the main contribution to the interaction comes from wavelengths in the vicinity of λc ∼ 4πL that can be smaller than the mean free path of electrons in a metal. In this case, the local dielectric function ε(ω) is not adequate for calculating the van der Waals interaction, and a nonlocal dielectric function is needed. In other words, one needs to use a frequency and wave vector dependent dielectric function ε(ω, q) . The physical meaning of having a spatially dispersive, or nonlocal, dielectric function is that the response of a material at point r to an external electric field depends on the value of the electric field at a neighboring point r0 . Usually, nonlocal effects are important when the wavelength of light is smaller than the mean free path of an electron in the material. Systems where the momentum of quasi-particles depends on wave vector, such as excitons in semiconductors, also exhibit a nonlocal dielectric function.3 The importance of nonlocality at the nanoscale can be seen in several recent studies. García de Abajo4,5 showed that inclusion of spatial dispersion in the optical response of noble metal nanoparticles produces a blue shift and broadening in the r 2011 American Chemical Society
plasmon resonance of spheres, shells, and metallic waveguides. Similarly, the work of Yannopapas6 included nonlocality in the optical response of a 2D array of metallic nanoparticles, also finding a blue shift in the optical spectra of the array. Temperature and nonlocal effects can change the polarizability of mesoscopic metallic nanoparticles, as shown by Chen7 in a comparison of different metallic spheres. McMahon et al.8,9 for the first time introduced the effects of spatial dispersion in computational methods for determining the optical properties of arbitrary shaped particles, away from the static limit. In addition to a blue shift present due to nonlocality, there is a suppression of the enhancement of the electric field compared to the local case. Recently, it was shown that unique resonances are present in plasmonic nanostructures not present in the local case.10 In the context of Casimir or retarded van der Waals forces, the role of spatial dispersion has also been considered in the Lifshitz formalism. Katz11 suggested the possible role of nonlocality and the need to quantify its effect. Later, Heinrichs12 studied the van der Waals interaction between two metallic plates, showing that at large separations spatial dispersion was not important. Podogornik13 studied the effect of spatial dispersion in the solvent, filling the space between two slabs described by a local dielectric function. Recently, the role of spatial dispersion in the accurate calculation of the Casimir force was studied.1420 In particular, the importance of including spatial effects at finite temperature was used to resolve recent controversies regarding the violation of Nerst’s heat theorem in Lifshitz theory.21,22 In this paper, we present a numerical study of the effects of spatial dispersion in the Hamaker coefficient and calculate the nonlocal van der Waals energy between nanoparticles to show that at small separations nonlocality has a significant effect in reducing the value of the van der Waals energy. Received: October 4, 2011 Revised: December 2, 2011 Published: December 05, 2011 420
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’ NONLOCAL OPTICAL RESPONSE In nonlocal media, the response to an electric field depends on time and position. The relation between the displacement D and the electric field E is Dðt, rÞ ¼
Z ∞
∞
dr0
Z t
∞
dt0 εðr r0 , t t0 ÞEðt0 , r0 Þ
ð1Þ
This means that the response at point Br depends on the value of the electric field at point r0 . In the local case, we have ε(r r0 , t t0 ) = ε(t t0 )δ(r r0 ) . By taking the Fourier transform of eq 1 of both space and time coordinates, the usual relation between vector displacement and electric field is obtained Dðω, qÞ ¼ εðω, qÞEðω, qÞ ð2Þ From Maxwell’s equations, in the nonlocal case we obtain the following two conditions24 εðω, qÞ 6¼ 0, q 3 E ¼ 0
ð3Þ
and εðω, qÞ ¼ 0, q E ¼ 0
ð4Þ
Equation 3 implies that the electric field is transverse, but in nonlocal media eq 4 implies that a longitudinal electric field can also be excited.23 This means that the dielectric function can be written as the sum of a transverse part εT(ω,q) and a longitudinal part εL(ω,q) as24 εðω, qÞ ¼ εL ðω, qÞ^eL þ εT ðω, qÞ^eT ð5Þ where ^eT,L are projectors along the transverse and longitudinal direction. In the limit q f 0, both longitudinal and transverse dielectric functions coincide. A challenge with nonlocal media is to model the dielectric function. A simple, yet effective, way of describing the dielectric function is using the hydrodynamic model.25,26 Although limited in its ability to describe phenomena such as Landau damping, it provides an accurate description of nonlocal optics. Some of the limitations of the hydrodynamic model have been overcome with Lindhard’s dielectric function. However, a modified hydrodynamic model that includes the corrections to describe a degenerate electron gas was proposed by Halevi.27 While keeping the simplicity of the hydrodynamic model, it incorporates the Mermin correction28 needed to relax the system to thermal equilibrium, and it is valid for arbitrary frequencies. In this corrected hydrodynamic model, the longitudinal dielectric function is given by ω2p ð6Þ εL ðω, qÞ ¼ 1 ωðω þ iγÞ β2 q2
’ BOUNDED SYSTEMS The nonlocal response defined in eq 1 is for an infinite space. However, the problem of extending the nonlocal dielectric theory to bounded systems has been widely considered in the literature since the pioneering work of Pekar.29 Experimentally, the influence of spatial dispersion on the optical properties of surfaces was reported by Lopez-Rios.3032 For a bounded medium, additional assumptions are needed to include nonlocal effects and to restore the broken translational symmetry due to the surface. The usual local approach to Fresnel optics considers only the transverse fields, and the effect of the surface is included in the usual boundary conditions derived from Maxwell’s equations. In the nonlocal case, the appearance of the longitudinal fields means that at the surface the transverse and longitudinal modes have to be coupled. To apply eq 1 to finite systems, we consider a nonlocal half-space whose surface is on the xy plane and which extends into z > 0 . The spatial integral over z in eq 1 is in the range z ∈ [0,∞]. To extend the integration range to z ∈ [∞,0], a typical approach by Kliewer and Fuchs is to introduce a fictitious or ancillary system for z < 0 which is a specular image of the real system.33 In other words, the fields for z < 0 have to satisfy the symmetry conditions Ex(z) = Ex(z) and Ez(z) = Ez(z). The ancillary system introduces an additional current density at the interface, thus an additional boundary condition has to be imposed. In this case, the current density normal to the surface has to vanish, i.e., Jz = 0 . Thus, a semi-infinite system is equivalent to an infinite one provided the fields for z < 0 are appropriately chosen.34 Consider, for example, the Fresnel coefficients that will be used again later in the paper. For a half-space of local dielectric function ε(ω) in contact with an external medium εm(ω), the Fresnel coefficients for p and s polarized light are
3ω=5 þ iγ=3 ω þ iγ
ω2p ωðω þ iγÞ
rs ¼
k km k þ km
ð9Þ ð10Þ
rp ¼
εT ðωÞkm εm ðωÞk þ Q 2 ðεT ðωÞ 1Þ=qL εT ðωÞkm þ εm ðωÞk Q 2 ðεT ðωÞ 1Þ=qL
ð11Þ
rs ¼
k km k þ km
ð12Þ
In this case, Q is the wave vector component parallel to the surface. For the hydrodynamic model, the s polarized Fresnel coefficient remains the same in the local and nonlocal cases. The nonlocal dielectric function enters in the evaluation of the longitudinal wave vector qL. The wave vector qL is the one that satisfies eq 4, i.e., ε(ω,qL) = 0, yielding
ð7Þ
with vf being the Fermi velocity. The transverse part of the dielectric function is the usual Drude type εT ðωÞ ¼ 1
εm ðωÞk εðωÞkm εm ðωÞk þ εðωÞkm
The wave vector component normal to the surface is k in the material and km in the external medium. If the half-space is now described by a nonlocal dielectric function ε(ω,q), the Fresnel coefficients for the hydrodynamic model are modified14
where ωp is the plasma frequency and γ is the damping parameter. The term β2, rather than being constant as in the usual hydrodynamic model, is frequency dependent β2 ¼ v2f
rp ¼
q2L ¼
ω2 ω2p þ iωγ β
ð13Þ
The important fact to notice is that nonlocal optical properties of a surface are not simply obtained by replacing a local dielectric function by a nonlocal one.35
ð8Þ 421
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’ HAMAKER COEFFICIENT. LOCAL AND NONLOCAL CASE Lifshitz theory predicts that the van der Waals interaction between two equal half spaces separated by a material with a local dielectric function εm(ω) of thickness L between them can be calculated using the Lifshitz theory of dispersive forces38 as U vdw ¼
kB T ∞ 2π n ¼ 0
∑
Z ∞ 0
ln½ð1 rp2 e2km L Þð1 rs2 e2km L ÞQ dQ ð14Þ
where T is the temperature; kB is Boltzman’s constant; and the prime in the summation means that the n = 0 term has to be multiplied by 1/2. It is important to recall that in Lifshitz’s theory the frequency axis was rotated to the complex axis. This means ω f iζn, where ξn = 2πkBTn/p are the Matsubara frequencies. This equation for the van der Waals interaction is general and can be used in the retarded and nonretarded limits and for the local and nonlocal media. The derivation of Lifshitz formula for the van der Waals interaction is based in Rytov’s theory of fluctuating electromagnetic fields36,37 that are solutions to Maxwell’s equations with fluctuating currents as sources. This is the reason the van der Waals energy depends on the Fresnel coefficients and is valid for the local and nonlocal cases. For the local case, eqs 9 and 10 will be used, and eqs 11 and 12 will be used for the nonlocal case. At the nanoscale, the nonretarded regime is of interest. By nonretardation it is meant that the interactions can be regarded as instantaneous and that the speed of light effectively is cf∞ . This means that the perpendicular components of the wave vectors are k ∼ Q and km ∼ Q; thus, for s-polarized light we have rs = 0, and for p-polarized light eq 9 and eq 11 become, respectively rnr ¼
εm ðωÞ εðωÞ ðlocalÞ εm ðωÞ þ εðωÞ
ð15Þ
rnr ¼
εT ðωÞ εm ðωÞ þ Q ðεT ðωÞ 1Þ=qL εT ðωÞkm þ εm ðωÞk Q ðεT ðωÞ 1Þ=qL
ð16Þ
Within the nonretarded approximation and introducing a new variable y = QL, the Lifshitz expression eq 14 for the van der Waals energy can be written in the Hamaker form39 A(L)/12πL2 as ! 1 3kB T ∞ 0 Z ∞ 2 y U vdw ¼ lnð1 rnr e Þydy ð17Þ 12π2 L2 2 n¼0 0
Figure 1. Hamaker coefficient between two Au half-spaces separated by water for the local and nonlocal cases.
Again, we see that the nonlocal case is not obtained by simply replacing the local dielectric function by the nonlocal ones.
’ HAMAKER COEFFICIENT BETWEEN TWO HALFSPACES For two parallel Au half-spaces, we calculate the Hamaker coefficient for the local and nonlocal cases within the hydrodynamic model. Between the slabs the space is filled with water. The dielectric function of water εm(ω) is modeled using a multiple Lorentz oscillator dielectric function.40 For Au, the dielectric function given by eqs 6 and 8 is used with ωp = 9 eV and γ = 0.035 eV. In the case of Au, as shown by Svetovoy,41 the parameters for the dielectric function vary depending on the optical data used, leading to several percent difference in the calculation of dispersive forces using Lifshitz formalism. In Figure 1 we show the Hamaker coefficient as a function of the separation L for the local and nonlocal dielectric functions. The local case shows the usual behavior as the separation decreases, with the value of the Hamaker coefficient being almost constant at small separations. However, the nonlocal case shows a surprisingly different behavior. As the separation between the slabs decreases, the Hamaker coefficient decreases by over an order of magnitude. Given the Hamaker coefficient, the van der Waals energy between the plates can be calculated as
∑
U ¼
where the term in parentheses is the Hamaker coefficient. Explicitly, for the local case we get the usual Hamaker coefficient ! 3kB T ∞ 0 Z ∞ εm ðωÞ εðωÞ 2 y ln 1 e Alocal ¼ 2 n¼0 0 εm ðωÞ þ εðωÞ ð18Þ and for the nonlocal case the Hamaker coefficient is
0
3kB T ∞ 0 2 n¼0
∑
Z ∞ 0
εT ðωÞ εm ðωÞ þ Q ðεT ðωÞ 1Þ=qL ln@1 T ε ðωÞkm þ εm ðωÞk Q ðεT ðωÞ 1Þ=qL
!2
ð20Þ
where L is the separation between the plates. In Figure 2 we show the van der Waals energy for two parallel plates for the local and nonlocal cases. The excitation of additional modes in the nonlocal case reduces the interaction energy. The nonlocal curve shows an inflection at a separation close to 0.1 nm. As the separation decreases, higher values of the wavevector components come into play in the longitudinal dielectric function, exciting longitudinal modes in the material, thus decreasing the interaction energy. From eq 6 we see that if the value of the parameter β increases, the position of the inflection will occur at larger separations. Of course, the value of β depends on the Fermi velocity and thus on the material being used, so the separation distance where nonlocal effects are expected to be important will depend on material. The differences between local and nonlocal results seen in Figure 2 will prevail for other geometries as will be shown later in the paper.
∑
Anonlocal ¼
A 12πL2
1 ey A
ð19Þ 422
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Figure 3. Local and nonlocal interaction between two Au spheres of radius R = 5 nm.
Figure 2. Local and nonlocal van der Waals energy between two Au half spaces.
’ VAN DER WAALS ENERGY BETWEEN NANOPARTICLES Finally, we explore the effects of nonlocality in the van der Waals interaction between nanospheres and between nanorods. These two geometries were chosen as examples of particles used in nanoassembly. Nanospheres, nanorods, and nanoprisms4244 are nowadays commonly synthesized and used. There are many procedures to change the sizes, shapes, and material properties45 of nanoscopic particles. The sizes and separations of these nanoparticles are in the range where nonlocal effects can be important. At small separations, the Hamaker approach provides a valuable tool for calculating the van der Waals interaction between particles of different shapes but using instead of the Hamaker constant the coefficient given by either eq 18 for the local case or eq 19 for the nonlocal case. For example, for two equal spheres of radius R, we have38 A R2 R2 þ 2 3 ð2R þ LÞ 4R 2 ð2R þ LÞ2 !! 1 4R 2 þ ln 1 2 ð2R þ LÞ2
Figure 4. Interaction energy between two Au parallel cylinders of radius R = 5.5 nm and length l = 34 nm. The dimensions correspond to the nanorods used in ref 47.
is shown. As in the case of the spheres, the nonlocal case shows a significant decrease as compared to the local calculation. For both cases of spheres and cylinders, the interaction energy shows the same qualitative differences between local and nonlocal results as in Figure 2. This is expected since in Hamaker’s approach the interaction energy uses the results for parallel slabs and then corrects for geometry. The case of spheres shows an actual minimum as a function of separation distance. Geometry alone does not cause this minimum; otherwise, it should be present for both the local and nonlocal cases.
U ss ¼
ð21Þ
where L is the separation between the surface of the spheres. We consider two equal Au spheres of radius R = 5 nm as the ones used by Wang46 to study the bistability and hysteresis of aggregation processes of charged nanoparticles. The interaction energy for the local and nonlocal case for the two spheres is shown in Figure 3. In the nonlocal case the interaction energy decreases as the separation between the spheres decreases. For large separation both local and nonlocal cases coincide. The other nanoparticle shape that has been used is nanorods. Walker47 used nanorods to achieve assembly mediated by charge-nduced dipole interactions. We approximate the nanorods as cylinders of radius R = 5.5 nm and length l = 34 nm. In this case, the van der Waals energy is U cc ¼
A Rl 48L3=2
ð22Þ
where L is the separation between the surface of the cylinders. In Figure 4 the local and nonlocal interaction between the nanorods
’ CONCLUSIONS We have shown that at the nanoscale the van der Waals interaction has to be modified. In a continuum description of the materials, the dielectric function that depends on frequency and wave vector has to be used in the Lifshitz formalism to capture the correct physical behavior of the dispersive forces. The effect of spatial dispersion is to decrease the value of the Hamaker coefficient at small separations. Using the nonlocal Hamaker coefficient, the van der Waals energy between two nanospheres or two nanorods was calculated. At small separations the difference between the local and nonlocal case can be as large as 2 orders of magnitude. In the presence of ionic solutions, the dielectric function of water is also described by a nonlocal dielectric function.13 A remaining open problem is the calculation of the Hamaker 423
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coefficient when both the materials and the medium between them are described by a nonlocal dielectric function. Another open issue relevant for nanoparticles is the length scales at which the Lifshitz continuum model works. Although there is no clearcut point where the theory stops being valid, other forces, such as Pauli forces, would become dominant at atomic separations, such that differences between different models of dispersive interactions would be of secondary importance. Inglesfield calculated the van der Waals interaction between two metal slabs using the coherent phase approximation (CPA), showing that at small separations, of the order of 0.5 nm, the difference between Lifshitz theory was about 10%48 Some advances in this direction have also been studied by Cole.49 However, the present results show that nonlocal effects are important in determining dispersive interactions at separations where Pauli repulsion effects are not important, which raises important questions concerning the adequacy of the DLVO theory at the nanoscale.
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’ AUTHOR INFORMATION Corresponding Author
*E-mail: raul@fisica.unam.mx.
’ ACKNOWLEDGMENT R.E.S. acknowledges the hospitality of the Non Equilibrium Energy Research Center of Northwestern University and the partial support of CONACyT project no. 82474. This research was supported by the NERC EFRC of the US DOE (BES Award de-sc0000989). Helpful discussions with members of the ESF Casimir Network and L. W. Mochan are acknowledged. ’ REFERENCES (1) Lifshitz, E. M. Zh. Eksp. Teor. Fiziol. 1955, 29, 894. Sov. Phys. JETP 1956, 2, 73. (2) Dzyaloshinskii, I. D.; Lifshitz, E. M.; Pitaevskii, L. P. Usp. Fiziol. Nauk 1961, 73, 381. Sov. Phys. Usp. 1961, 4, 153. (3) Flores-Desirena, B.; Perez-Rodríguez, F.; Halevi, P. Phys. Rev. B 1994, 50, 5404. (4) García de Abajo, F. J. Phys. Chem C 2008, 112, 17983. (5) David, C.; García de Abajo, F. J. J. Phys. Chem. C 2011, 115, 19470–19475. (6) Yannopapas, V. J. Phys.: Condens. Matter 2008, 20, 325211. (7) Chen, C. W.; Liao, L. S.; Chiang, H. P.; Leung, P. T. Appl. Phys. B: Laser Opt. 2010, 99, 223. (8) McMahon, J. M.; Gray, S. K.; Schatz, G. C. Phys. Rev. Lett. 2009, 103, 097403. (9) McMahon, J. M.; Gray, S. K.; Schatz, G. C. Phys. Rev. B 2010, 82, 035423. (10) Raza, S.; Toscano, G.; Jauho, A.; Wubs, M.; Asger Mortensen, N. Phys. Rev. B 2011, 84, 121412 (R). (11) Katz, E. I. Sov. Phys. JETP 1977, 46, 109. (12) Heinrichs, J. Phys. Rev. B 1975, 11, 3625. (13) Podogornik, R.; Cevc, G.; Zeks, B. J. Chem. Phys. 1987, 87, 5957. (14) Esquivel-Sirvent, R.; Villarreal, C.; Mochan, W. L. Phys. Rev. A 2003, 68, 052103. (15) Esquivel-Sirvent, R.; Villarreal, C.; Mochan, W. L. Phys. Rev. A 2005, 71, 029904. (16) Esquivel-Sirvent, R.; Mochan, W. L. Quantum Field Theory under the Influence of External Conditions; Milton, K., Ed.; Rinton Press: Princeton, NJ, 2004; p 90. (17) Esquivel, R.; Svetovoy, V. B. Phys. Rev. A 2004, 69, 062102. 424
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