Article pubs.acs.org/JPCC
Size-Dependent Hamaker Constant for Silver Nanoparticles Anatoliy O. Pinchuk Department of Physics and Energy Science, University of Colorado at Colorado Springs, 1420 Austin Bluffs Pkwy, Colorado Springs, Colorado 80933, United States ABSTRACT: The nonretarded size-dependent Hamaker constant for spherical silver nanoparticles was calculated by using the Lifshitz theory. The size dependence of the Hamaker constant was taken into account based on the size effect of the dielectric permittivity of silver nanoparticles. The experimental data for bulk silver dielectric permittivity was modified to include the intrinsic size effect of the scattering of the conduction electrons by the surface of the nanoparticles. The intrinsic size effect leads to a substantial increase of the Hamaker constant for nanoparticles with radii of the order 1 nm, when the mean free path of conducting electrons in the nanoparticles is limited by the particles’ boundaries.
1. INTRODUCTION Metal nanoparticles exhibit unique optical and electronic properties caused by collective electronic excitations of conduction electrons known as surface plasmon resonance (SPR).1,2 These coherent electronic excitations lead to strong absorption of light at SPR wavelength and result in a large enhancement of the local electric field close to the surface of the nanoparticles.3,4 These unique properties can be used in numerous applications of metal nanoparticles in such areas as surface enhanced spectroscopy, for example, surface enhanced Raman scattering (SERS),5−8 medical diagnosis and cancer treatment,9,10 photovoltaic,11 and near-field microscopy,12 to name just a few. Metal nanoparticles have been studied to develop subwavelength optical waveguides,13,14 subwavelength microscopy, and optical biochemical sensors.15−17 Self-assembly of metal nanoparticles has recently been studied as a potentially powerful bottom-up fabrication technique. With current resolution limitations of the traditional photolithography and electron-beam lithography and their high costs, self-assembly of nanoparticles might pave an alternative route to low cost fabrication of nanostructures. Recent reports on nanoparticle self-assembly using DNA linkers,18 electrostatic,19,18 dipole−dipole,20 and dispersion interactions21,22 between nanoparticles indicate the potential of this approach.23,24 Understanding the interaction between nanoparticles at the nanoscale is essential for the precise control over the self-assembly process. The van der Waals interaction is one of the major components of the interaction forces between nanoparticles in a solution.25 The van der Waals potential is usually split into a product of a size- and morphology-independent constant, known as the Hamaker or Lifshitz constant, and a sizedependent term.25 Thus, the Hamaker constant has been considered so far as a morphology- and size-independent constant that depends only on the dielectric properties of the constituent materials.25 Noble metal nanoparticles exhibit a socalled size effect,1,26 which influences their extinction proper© 2012 American Chemical Society
ties: as the size of the nanoparticles becomes smaller, their SPR shifts and broadens.1,27,28 In addition, the size effect leads to modification of the dielectric permittivity of the nanoparticles.1,3,26 The Hamaker constant is intrinsically connected to the absorption properties of the interacting objects,25 for example, nanoparticles, and thus, it depends on the size and morphology of the nanoparticles. The Hamaker constants are known for many materials, such as oxides, polymers, and dielectrics, with high accuracy whereas for noble metals these constants span through a larger range: 90−300 zJ (1 zJ = 10−21 J) for gold across water25 and 100− 400 zJ for silver across water.25 The origin of the large uncertainties in the magnitude of these constants can be caused by the intrinsic size effect in the nanoparticles. The effect of nonlocal dielectric function on the Hamaker constant was recently shown for small particles.35 The frequency-dependent damping was discussed in details in the context of the Casimir force between two gold films.37 However, the influence of the internal size effect on the Hamaker constant was not discussed so far. In this paper, we show theoretically the size dependence of the Hamaker constant for silver nanoparticles. We use the Lifshitz theory of dispersion forces to calculate the Hamaker constant for silver nanoparticles with the intrinsic size effect taken explicitly into account.
2. HAMAKER AND LIFSHITZ CONSTANTS van der Waals forces, also known as dispersion forces, result in weak interaction between nanoparticles in colloidal solutions.31 These forces depend on the materials from which the nanoparticles are made through a so-called Hamaker or Lifshitz constant.31 In this paper, we analyze the size dependence of the Hamaker constant for spherical silver nanoparticles whose radii Received: June 22, 2012 Revised: August 10, 2012 Published: August 29, 2012 20099
dx.doi.org/10.1021/jp3061784 | J. Phys. Chem. C 2012, 116, 20099−20102
The Journal of Physical Chemistry C
Article
vary around R ∼ 10 nm. We assume that the nanoparticles are located either in vacuum or in water. Let us consider the interaction between two spherical particles each having permittivity ε1 and ε2 and immersed in a medium with permittivity ε3. We assume the particles have different radii Ri (i = 1, 2). Both radii are supposed to be much larger than the surface-to-surface separation d (Ri ≫ d), see Figure 1a. The nonretarded Hamaker constant A132 in this case
3. SIZE-DEPENDENT HAMAKER CONSTANT FOR METAL NANOPARTICLES The size dependence of the dielectric permittivity of the interacting nanoparticles leads to the size dependence of the Hamaker constants. The dielectric permittivity for metal nanoparticles depends on the size, known as the internal size effect.1,3 The bulk dielectric permittivity for metal nanoparticles can be modified to include the internal size effect for nanoparticles with the size smaller than the electron mean free path in the bulk.26 In our calculations, we used εAg(ω) and εH2O(ω) as given in the literature34 based on experimental data for optical constants of silver and water. The εAg(ω) experimental data were modified to include the size effect. The scattering of the electrons from the surface of the nanoparticles as well as interface of chemical damping effects26 can be accounted for by replacing the bulk dielectric constant with the size-modified dielectric permittivity, εAg(ω):
Figure 1. Geometry of the system under consideration.
εAg(ω) → εAg(ω , R ) = εAg(ω) + Δε(ω , R )
where
can be calculated based on the Lifshitz theory30,31 of the van der Waals forces between two parallel semi-infinite plates separated by the distance d, Figure 1b. The Lifshitz theory gives the expression for the Hamaker constant:31,32 A132
3 = ℏω132 4π
Δε(ω , R ) = εDrude(ω , R ) − εDrude(ω)
εDrude(ω) = 1 −
(1)
where ω132 is the Lifshitz constant defined as
ωp2 ω 2 + i γω
εDrude(ω , R ) = 1 −
∞
ω132
∫0
∞
∫ 0
x2 [ε1(iω) + ε3(iω)][ε2(iω) + ε3(iω)] x e [ε1(iω) − ε3(iω)][ε2(iω) − ε3(iω)]
dx dω −1
2 π
∫0
∞
x Im εn(x) x 2 + ω2
dx
(3)
The expression for the Lifshitz constant can be obtained by evaluating the integral31,32 in eq 2: ω132 ≈
∫0
∞
ωp2 ω 2 + iγ(R )ω
(8)
where vF is the Fermi velocity of the metal, R is the radius of the metal nanoparticle, and A is a dimensionless constant of the order of 1.1,3 The parameters in eqs 7−9 were taken from the literature:3,34
[ε1(iω) − ε3(iω)][ε2(iω) − ε3(iω)] dω [ε1(iω) + ε3(iω)][ε2(iω) + ε3(iω)]
≡ ω̃132
(7)
are the Drude permittivity and the size-modified Drude permittivity of the metal, with ωp, γ, and γ(R) being, respectively, the plasma frequency, electron scattering rate or damping, and the modified (size-dependent) scattering rate of the metal. The latter is defined as1,3,26 v γ (R ) = γ + A F (9) R
(2)
where εn(iω) is the permittivity of the nth medium as a function of the imaginary frequency iω, εn(iω) = 1 +
(6)
is the modification of the bulk dielectric permittivity εAg(ω) due to the size effect, and
31
1 = 2
(5)
ωp = 5.8 × 1015 s−1, γ = 0.01ωp , vF = 1.39 × 106 m/s,
(4)
(10)
A=1
In what follows, we consider silver nanoparticles located in vacuum and in water. We use the nonretarded symmetric Hamaker constants A1v1 and A1w1, where the indices “v” and “w” denote the embedding medium (either vacuum or water), and the permittivity εn(ω), ε1(ω) = ε2(ω) = εAg(ω), and ε3(ω) = 1 (vacuum) or ε3(ω) = εH2O(ω), respectively.
According to eqs 1−3, the Hamaker constant is defined by the values of the permittivity εn(ω) over the entire frequency range 0 < ω < ∞. However the experimental data for εAg(ω) and εH2O(ω) are only available within the limited frequency range. The data that we used in our calculations for εAg(ω)
Table 1. Calculated Hamaker Constant Values A131 (zJ) for Silver Nanoparticles in Vacuum and Water
size-corrected dielectric function εAg(ω, R) for different R (nm) system
εAg(ω) no size effect in calculations
1
5
10
50
100
Ag−vacuum Ag−H2O
140.6 33.54
234.2 79.82
208.7 61.67
186.3 52.62
155.8 39.96
149.5 37.26
20100
dx.doi.org/10.1021/jp3061784 | J. Phys. Chem. C 2012, 116, 20099−20102
The Journal of Physical Chemistry C
Article
ω131 and A131. Therefore, the size effect results in an increase in ε″Ag, εAg(iω), ω131, and A131. The physics behind this effect can be rationalized by the dispersive nature of van der Waals forces. Thermal fluctuations of the electric dipole moment induce a dipole moment in a closely neighboring nanoparticle, and energy minimization of the two dipoles system results in an attractive van der Waals force. The two dipoles will attract each other with a stronger force if their dipole moments have the same resonance frequency or if they have larger absorption cross sections. The internal size effect broadens the absorption band in small nanoparticles, which leads to effective absorption of electromagnetic waves. Since the absorption is stronger in small nanoparticles, their interaction, in this case attraction, becomes stronger, that is, the Hamaker constant becomes larger for small nanoparticles. Let us now compare our calculations with the size effect with those previously reported in the literature. Note, however, that the reported Hamaker constants for silver across vacuum or water have different values depending on the method used for their calculation; sometimes the difference is about a factor of 16 (17−282 zJ).31,32 Therefore, we will compare our results with those obtained by the same method. Table 2 provides the
include the IR to X-ray ranges with the wavelength spanning through λ ≈ 1 Å up to 10 μm,34 while the data for εH2O(ω) span the IR to UV frequency range with the wavelength λ ≈ 200 nm to 200 μm.33 The integrals in eqs 2−4 were numerically integrated within the available frequency range of the experimental data. When calculating the Hamaker constant A1v1 for silver across vacuum, we performed integration up until ωmax = 10 × 1015, 50 × 1015, and 100 × 1015 s−1, all within the frequency range of the experimental data for εAg(ω). The calculations of the constant A1w1 for silver across water were performed up to the maximum frequency ωmax to be 9.4 × 1015s−1, which is the maximum available experimental frequency for εH2O(ω) data. The numerical results corresponding to the nonretarded symmetric Hamaker constants for Ag nanoparticles across vacuum and water are given in Table 1. We used a linear interpolation of the experimental data of εAg(ω) and εH2O(ω) for both Ag−vacuum and Ag−water systems. The integration was done in eq 2 up until ωmax = 50.0 × 1015 s−1, while for the Ag−water system, ωmax = 9.4 × 1015 s−1. The second column shows calculated results of the Hamaker constant for two interacting systems with the bulk dielectric permittivity for silver taken from the experimental published results, that is, no size effect was included in the dielectric permittivity εAg(ω). The next five columns present data for the Hamaker constant for the same two interacting systems (Ag−vacuum and Ag− H2O) but with the dielectric permittivity εAg(ω, R) corrected for the size effect as outlined in eqs 5−9.
Table 2. Comparison of the Lifshitz Constant Values ℏω131 (eV) Calculated from the Exact and Approximate Expressions (No Size Effects) Krupp et al.29 system
4. DISCUSSION Our calculations reveal that the size effect leads to increased values of the Hamaker constants. This effect can be rationalized by the increase of the imaginary part of permittivity of metal nanoparticles because of the internal size effect. The imaginary part of the Drude permittivity ″ (ω) = εDrude
(11)
(12)
where we used the fact γ ≪ ωp. The size effect results in, according to eq 9, increase of the damping factor γ and thus an increase of ε″Drude(ω), eq 12. The same is true not only for the Drude permittivity but also for the size-modified experimental permittivity of silver, eq 5. Its imaginary part εAg ″ contributes via eq 3 to the function εAg(iω), and so, the larger εAg ″ is, the larger εAg(iω) will be. The latter in turn contributes to the Lifshitz constant via (eq 2) or in a simpler way via (eq 4). In the symmetric case we consider here, the integrand in eq 4 can be presented in the form ⎛ ⎞2 2ε3(iω) = − 1 ⎜ ⎟ ε1(iω) + ε3(iω) ⎠ [ε1(iω) + ε3(iω)]2 ⎝
approximate
Ag−H2Ob
not available
3.04 3.68b 0.878
4.90 9.46b 0.855
comparison of our results for the Lifshitz constant ω131 with those obtained by Krupp et al.29 All ω131 data have been calculated without taking into account the size effect by using eq 4 as an approximate number and eq 2 as an “exact” number. Note that Krupp et al. did not use the “exact” expression (eq 2) for the Lifshitz constant and thus provided only results based on approximate eq 4. Also, they did not provide the Lifshitz constant value for the Ag−H2O system. Nevertheless, from the comparison of the approximate values for the Ag−vacuum system, one can conclude that our results agree reasonably well with those due to Krupp et al.29 The results presented in Tables 1 and 2 were obtained by using a linear interpolation of the experimental spectral data for both silver and water. The use of other types of the interpolation changes the results slightly, which is illustrated in Table 3. Here, all the Hamaker constants were calculated by using eq 2 with taking into account the size effect with R = 5
ωp2γ ω3
exact
10.41
ωmax = 38.0 × 1015 s−1. bωmax = 50.0 × 1015 s−1. cωmax = 9.4 × 1015 s−1.
at high enough frequencies ω ≥ ωp (that contribute substantially to the Lifshitz and Hamaker constants) has the asymptotic form ″ (ω) = εDrude
approximate
Ag−vacuuma
a
ωp2γ ω3 + γ 2ω
our results
Table 3. Influence of the Interpolation Used for the Spectral Data on the Hamaker Constant Value A131 (zJ) at R = 5 nm
[ε1(iω) − ε3(iω)]2
interpolation method
(13)
where ε1(iω) ≡ εAg(iω). Since εn(iω) are always positive and monotonic functions,31 it is obvious from eq 13 that the larger εAg(iω) is, the larger the integrand and consequently the larger
a
20101
system
linear
cubic
spline
Ag−vacuuma Ag−H2Ob
208.7 61.67
208.6 61.60
208.6 61.59
ωmax = 50.0 × 1015 s−1. bωmax = 9.4 × 1015 s−1. dx.doi.org/10.1021/jp3061784 | J. Phys. Chem. C 2012, 116, 20099−20102
The Journal of Physical Chemistry C
Article
nm, ωp = 5.8 × 1015 s−1, and γ = 0.01ωp. The extrapolation of the experimental data for silver into far IR and UV wavelength regions36 might slightly change the numerical results and will be discussed in details in an upcoming publication. Finally, note that the results on the size effect on the Hamaker constant for silver nanoparticles are valid in general for metal nanoparticles that exhibit a similar size effect in their optical and dielectric response, for example, gold or copper nanoparticles.1,26,27 The calculations of the corrections in the Hamaker constant for other metal nanoparticles should follow the theory outlined in eqs 1−9. The experimental data for the damping constant that is calculated from the optical response of such nanoparticles should be taken from the results available in the literature.34
(13) Salerno, M.; Krenn, J.; Hohenau, A.; Ditlbacher, H.; Schider, G.; Leitner, A.; Aussenegg, F. Opt. Commun. 2005, 248, 543−545. (14) Steinberger, B.; Hohenau, A.; Ditlbacher, H.; Aussenegg, F.; Leitner, A.; Krenn, J. Appl. Phys. Lett. 2007, 91, 081111. (15) Willets, K.; Van Duyne, R. Annu. Rev. Phys. Chem. 2007, 58, 267−272. (16) Yonzon, C.; Zhang, X.; Zhao, J.; Van Duyne, R. Spectroscopy 2007, 22, 42−47. (17) Zhao, J.; Zhang, X.; Yonzon, C.; Haes, A.; Van Duyne, R. Nanomedicine 2006, 1, 219−225. (18) Park, S.; Lytton-Jean, A.; Lee, B.; Weigand, S.; Schatz, G.; Mirkin, C. Nature 2008, 451, 553. (19) Kalsin, A.; Fialkowski, M.; Paszewski, M.; Smoukov, S.; Bishop, K.; Grzybowski, B. Science 2006, 312, 420−424. (20) Talapin, D. V.; Shevchenko, E. V.; Murray, C. B.; Titov, A. V.; Kral, P. Nano Lett. 2007, 7, 1213−1217. (21) Schmid, G.; Simon, U. Chem. Commun. 2005, 697−699. (22) Shevchenko, E. V.; Talapin, D. V.; Kotov, N. A.; O’Brien, S.; Murray, C. B. Nature 2006, 439, 55−56. (23) Kalsin, A.; Pinchuk, A.; Smoukov, S.; Paszewski, M.; Schatz, G.; Grzybowski, B. Nano Lett. 2006, 6, 1896−1903. (24) Grzybowski, B.; Winkleman, A.; Wiles, J.; Brumer, Y.; Whitesides, G. Nat. Mater. 2003, 2, 241−245. (25) Parsegian, V. A. Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists; Cambridge University Press: New York, 2006. (26) Pinchuk, A.; Kreibig, U. New J. Phys. 2003, 5, 151. (27) Reinholdt, A.; Pecenka, R.; Pinchuk, A.; Runte, S.; Stepanov, A.; Weirich, T.; Kreibig, U. Eur. Phys. J. D 2004, 31, 69−72. (28) Pinchuk, A.; Kreibig, U.; Hilger, A. Surf. Sci. 2004, 557, 269− 280. (29) Krupp, H.; Schnabel, W.; Walter, G. J. Colloid Interface Sci. 1972, 39, 421−427. (30) Lifshitz, E. M. Sov. Phys. − JETP 1956, 2, 73−82. (31) Barash, Yu. S. Van der Waals Forces; Nauka: Moscow, Russia, 1988 (in Russian). (32) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press, London, 1992. (33) Hale, J. M.; Querry, M. R. Appl. Opt. 1973, 12, 555−558. (34) Palik, E. D. Handbook of Optical Constants of Solids; Academic Press: Orlando, FL, 1985. (35) Esquivel-Sirvent, R.; Schatz, G. C. J. Phys. Chem. C 2012, 116, 420−424. (36) Svetovoy, V. B.; van Zwol, P. J.; Palasantzas, G.; De Hosson, J.; Th., M. Phys. Rev. B 2008, 77 (035439), 1−12. (37) Esquivel-Sirvent, R. J. Phys.: Conf. Ser. 2009, 161 (012012), 1−5.
5. CONCLUSION The Hamaker and Lifshitz constants for small silver nanoparticles with the size smaller than the electron free path in bulk metal depend on the size of the nanoparticles. The numerical calculations by using Lifshitz integrals reveal that the sizecorrected constant can be as much as 138% larger as compared to calculations without the size-corrected dielectric function of silver nanoparticles. The size effect in silver nanoparticles is related to the internal size effect caused by the scattering of conduction electrons from the surface of the nanoparticles. The increased scattering rate results in larger damping and larger imaginary part of the complex size-dependent dielectric permittivity of silver. This size dependence in turn leads to larger integrands in the Lifshitz integrals and larger Lifshitz and Hamaker constants.
■
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS The author acknowledges financial support from the NSF (grant no. DMR-0907614). The author thanks V. V. Gozhenko and J. L. Vedral for fruitful discussions and some help with numerical calculations.
■
REFERENCES
(1) Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer: Berlin, Germany, 1995. (2) Genet, C.; Ebbesen, T. Nature 2007, 445, 39−42. (3) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley: New York, 1998. (4) Barnes, W.; Dereux, A.; Ebbesen, T. Nature 2003, 424, 824−825. (5) Dieringer, J.; Lettan, R.; Scheidt, K.; Van Duyne, R. J. Am. Chem. Soc. 2007, 129, 16249−16253. (6) Whitney, A.; Elam, J.; Stair, P.; Van Duyne, R. J. Phys. Chem. C 2007, 111, 16827−16833. (7) Zhao, L.; Jensen, L.; Schatz, G. Nano Lett. 2006, 6, 1229−1234. (8) Laurent, G.; Felidj, N.; Grand, J.; Aubard, J.; Levi, G.; Hohenau, A.; Krenn, J.; Aussenegg, F. J. Microsc. (Oxford, U. K.) 2008, 229, 189− 196. (9) Hirsch, L.; Gobin, A.; Lowery, A.; Tam, F.; Drezek, R.; Halas, N.; West, J. Ann. Biomed. Eng. 2006, 34, 15−18. (10) Lowery, A.; Gobin, A.; Day, E.; Halas, N.; West, J. Int. J. Nanomed. 2006, 1, 149−154. (11) Cole, J.; Halas, N. Appl. Phys. Lett. 2006, 89, 153120−153125. (12) Drezet, A.; Hohenau, A.; Krenn, J. Phys. Rev. A 2006, 73, 062112−062117. 20102
dx.doi.org/10.1021/jp3061784 | J. Phys. Chem. C 2012, 116, 20099−20102
