Damping by Bulk and Shear Viscosity of Confined Acoustic Phonons

Jun 5, 2007 - A nanoparticle in aqueous solution is modeled as a homogeneous elastic isotropic continuum sphere in contact with an infinite viscous ...
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J. Phys. Chem. B 2007, 111, 7457-7461

7457

Damping by Bulk and Shear Viscosity of Confined Acoustic Phonons for Nanostructures in Aqueous Solution Lucien Saviot* Institut Carnot de Bourgogne, UMR 5209 CNRS-UniVersite´ de Bourgogne, 9 AV. A. SaVary, BP 47 870, F-21078 Dijon Cedex, France

Caleb H. Netting and Daniel B. Murray* Mathematics, Statistics and Physics Unit, UniVersity of British Columbia Okanagan, 3333 UniVersity Way, Kelowna, British Columbia, Canada V1V 1V7 ReceiVed: March 5, 2007; In Final Form: April 10, 2007

A nanoparticle in aqueous solution is modeled as a homogeneous elastic isotropic continuum sphere in contact with an infinite viscous compressible Newtonian fluid. The frequencies and damping of the confined vibrational modes of the sphere are calculated for the material parameters of a CdSe nanoparticle in water and a poly(methyl methacrylate) nanosphere in water. Although the effects of viscosity are found to be negligible for macroscopic objects, for nanoscale objects, both the frequency and damping of the vibrational modes are significantly affected by the viscosity of the liquid. Furthermore, both shear viscosity and bulk viscosity play an important role. A model of the spherical satellite tobacco mosaic virus consisting of outer solid layers with a water core is also investigated, and the viscosity of the water core is found to significantly damp the free vibrational modes. The same approach can be applied for nonspherical geometries and also to viscoelastic nanoparticles.

1. Introduction Confined acoustic phonons in nanostructures are mechanical vibrations whose frequency is on the order of the speed of sound divided by a typical length of the structure. For a nanoparticle embedded within a solid material, vibrational energy is gradually lost due to radiation of sound waves into the surrounding matrix, resulting in damped modes. Nanoparticles embedded in a solid transparent material can have their vibrations observed by Raman scattering or pulsed laser experiments. Nanoparticles in a liquid can have their vibrations observed by pulsed laser experiments1 and also by light scattering techniques.2 Nanoparticles have typical diameters from several nm to one hundred nm and thus have many atoms. For the lowest frequency acoustic phonons, the wavelength is long enough compared to the atomic spacing so that it is a useful approximation to consider the nanoparticle as a homogeneous continuum sphere. Furthermore, it is frequently convenient to assume isotropic elastic properties. The classical mechanical problem of finding the vibrational frequencies of a free, isotropic, homogeneous, continuum sphere was first solved by Lamb.3 The only parameters required are the sphere radius, density, and speeds of sound in the bulk material. For a nanoparticle embedded in a solid matrix, it is useful to idealize it as a continuum sphere in contact with an infinite continuum matrix. This problem was solved by Dubrovskiy.4 The resulting mode frequencies are complex numbers. The nonzero imaginary part reflects the fact that the oscillatory motion is decaying in time, since energy is flowing away from the vibrating nanoparticle, in sound waves moving away in the * To whom correspondence should be addressed. E-mail: lucien.saviot@ u-bourgogne.fr (L.S.); [email protected] (D.B.M.).

matrix. The only parameters required are the densities of the nanoparticle and the matrix, along with the sound speeds in the nanoparticle and the matrix. The situation of a nanoparticle embedded in a liquid medium could plausibly be approached as a special case of the Dubrovskiy solution. Liquids have zero transverse sound speed. However, a physical property of the liquid that is ignored in such a calculation is the viscosity. Here, we report the first theoretical calculation of the vibrational frequencies of a solid sphere embedded in a viscous, compressible liquid. Specifically, we find that, given the bulk and shear viscosities of water and other common solvents, the effects of viscosity are significant for all structures with nanoscale dimensions (100 nm or smaller). However, these effects are negligible for microscale and larger structures. Initial interest in the vibrational modes of viruses was motivated by the possibility of using ultrasound waves in resonance with these modes in order to destroy viruses in a living host.5 There is a report of such a mode being observed in M13 viruses using Raman scattering.6 A rough estimate of the frequency of these modes was initially made by considering the virus as a free elastic sphere while ignoring the effect of the liquid solvent.5 The effect of the surrounding water on the vibrational modes of a virus (or related biological structure) was considered later.7-10 However, these works all treated water as an elastic medium with zero transverse speed of sound and ignored the viscosity. The appearance of the effects of viscosity on nanoscale vibrations has previously been pointed out for the situation of a liquid drop as a model for a virus.5 Viscosity of the solvent was explicitly taken into account but only for torsional modes in an early article pointing out the relevance of the study of vibrations for biological structures.11

10.1021/jp071765x CCC: $37.00 © 2007 American Chemical Society Published on Web 06/05/2007

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Saviot et al.

2. Sphere in Inviscid Liquid In this section, we derive the vibrational modes of a solid sphere in a nonviscous fluid. This problem corresponds to the limit of zero transverse matrix sound speed in the Dubrovskiy solution. Consider the sphere to be of density Fp with isotropic elasticity so that its longitudinal and transverse speeds of sound are VLp and VTp, respectively. The density of the liquid matrix is Fm and the speed of longitudinal sound waves in the liquid is VLm. VTm is zero. The displacement field is represented by b u. As for any vector field, b u can be expressed as the sum of (1) a zero curl part and (2) a zero divergence part as follows:

b u ) ∇Φ + ∇ × Ξ B

(1)

The vector field Ξ B can be expressed in terms of two scalar fields ψ and χ

Ξ B ) ∇ × ∇ × (bχ) r + ∇ × (bψ) r

(2)

Vibrational modes can be classified as torsional (TOR) or spheroidal (SPH). TOR modes involve only the potential χ. SPH modes involve only the potentials Φ and ψ. In the case of the SPH modes which are breathing modes (where only radial motion is involved), only the Φ potential is needed. Modes can be furthermore classified by the integer angular momentum number l g 0 and the angular momentum zcomponent m where -l e m e l. There is no variation of frequency with m. Therefore, in what follows we consider the m ) 0 case only. The form of Φ is

Φ ) gl(kLr)Pl(cos θ)

(3)

where the function gl could be any of four types of Bessel (2) function: jl, nl, h(1) l , or hl . For clarity, we mention the l ) 0 cases of these functions: sin (z)/z, -cos(z)/z, -ieiz/z, and ie-iz/ z, respectively. Pl is the Legendre polynomial of order l. kL is the wavevector of longitudinal waves, where kL ) ω/VL. The forms of ψ and χ are

ψ ) gl(kTr)Pl(cos θ)

(4)

χ ) gl(kTr)Pl(cos θ)

(5)

where kT is the wavevector of transverse waves, where kT ) ω/VT. The boundary conditions which apply at the surface of the sphere are related to the stress within the materials. We begin with the 3 by 3 strain tensor, ij, which is

ij )

(

)

1 ∂ui ∂uj + 2 ∂xj ∂xi

(6)

The 3 × 3 stress tensor, σij, is related to the strain as follows:

σij ) 2µLij + λLkkδij

(7)

where µL and λL are the Lame´ constants, related to the speeds of sound via VT ) (µL/F)1/2 and VL ) [(λL + 2µL)/F]1/2. The bulk modulus is given by B ) λL + (2/3)µL. The summation over k is implicit. δij is the Kronecker delta. To obtain the mode frequencies, the three classes of vibrational modes need to be treated as separate cases.

Case 1: Torsional. For these modes, the displacement is purely along the φ direction, so that the external shape of the nanoparticle is unaffected by the vibration. The displacement field within the nanoparticle comes from the potential

χ ) Ajl(kTpr)Pl(cos θ)

(8)

whereas outside the nanoparticle the liquid is not affected. At the surface, the rφ component of the stress tensor vanishes: σrφ(in) ) 0. This boundary condition yields realvalued frequencies ω corresponding to undamped vibrational modes. Case 2: Spheroidal. For these modes, the displacement is along both the θ and r directions. Both the curl and divergence of the displacement field are nonzero. The displacement field within the nanoparticle comes from the two potentials

Φ ) Ajl(kLpr)Pl(cos θ)

(9)

ψ ) Bjl(kTpr)Pl(cos θ)

(10)

whereas outside the nanoparticle it is

Φ ) Ch(2) l (kLmr)Pl(cos θ)

(11)

The selection of h(2) in the matrix, rather than h(1) l l , is determined by our aesthetic wish that the imaginary parts of mode frequencies be positive. Since sound energy is flowing away from the vibrating nanoparticle, sound waves in the matrix are moving only in the +r direction and must be of the form f(r - Vt). Since the time dependence is eiωt, the r dependence must include a factor of e-ikr. At the nanoparticle surface, ur(in) ) ur(out). Also, σrr(in) ) σrr(out). Finally, σrθ(in) ) 0. The resulting 3 by 3 homogeneous system of equations (in A, B, and C) has a solution only at complex-valued frequencies, ω, corresponding to damped vibrational modes. Case 3: Breathing. For these modes, the displacement is purely radial. The displacement field within the nanoparticle is the gradient of the potential

Φ ) Aj0(kLpr)P0(cos θ)

(12)

whereas outside the nanoparticle the potential is

Φ ) Bh(2) 0 (kLmr)P0(cos θ)

(13)

The boundary conditions are ur(in) ) ur(out) and σrr(in) ) σrr(out). The resulting 2 × 2 homogeneous system of equations in A and B has a solution only at complex-valued frequencies, ω, corresponding to vibrational modes. 3. Viscous Compressible Fluid Consider a homogeneous isotropic medium with density F, displacement field b u, stress tensor σij, and strain tensor ij. For a solid material without any energy dissipation mechanism, the relationship between stress and strain is given by eq 7. Alternatively, for a purely viscous liquid medium we can write

σij ) 2µ

d du b  + λ∇‚ δij dt ij dt

(14)

where µ is the shear viscosity and λ is the second viscosity coefficient. The bulk viscosity is defined as ζ ) λ + (2/3)µ.

Damping of Confined Acoustic Phonons

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TABLE 1: Material Constants Used for Figures 1-6 F VL VT µ ζ

kg/m3 m/s m/s Pa‚s Pa‚s

CdSe

H2O

PMMA

5810 3570 1540 0 0

1000 1498 0 0.00089 0.0021512

1180 2800 1400 0 0

Now, we combine eqs 7 and 14 for stress by considering a viscoelastic (Kelvin-Voigt) material. Furthermore, we assume that the time variation of b u has frequency ω (going as eiωt) so that each occurrence of d/dt is replaced with iω. In that case, the combined expression for the stress is

σij ) 2(µL + iωµ)ij + (λL + iωλ)∇‚u bδij

(15)

The displacement field b u can be separated into a curl-free part and a divergence-free part. The solutions of the equation of motion for the curl-free part are longitudinal waves moving at speed

VL )

x

λL + iωλ + 2(µL + iωµ) F

(16)

whereas the solutions of the equation of motion for the divergence-free part are transverse waves moving at speed

VT )

x

µL + iωµ F

(17)

The above may be applied to the problem of a solid elastic vibrating sphere immersed in a viscous liquid. The example we have in mind is a spherical nanoparticle in water. The shear modulus, µL, of water is zero. Its bulk modulus is obtained from its long wavelength sound speed. The shear viscosity and bulk viscosity of water are also both known at various pressures and temperatures. The formalism already exists to calculate the vibrational frequencies of a solid sphere embedded in a solid matrix.4 This approach is simply extended to the case of an object in water by using the longitudinal and transverse speeds of sounds from eqs 16 and 17 as if they are the speeds of sound of a solid matrix. Because the real and imaginary parts of these speeds increase with ω, we expect viscosity to be more and more important as the typical length of the object decreases. 4. Numerical Results We have calculated mode frequencies for two illustrative cases of a nanoparticle in a liquid: First CdSe in water and second poly(methyl methacrylate) (PMMA) in water. In the first example, the material of the nanoparticle is extremely hard relative to the compressibility of the water. In the second example, the nanoparticle is only slightly “harder” than the liquid. The values of the material constants used are presented in Table 1. The value of the bulk viscosity for water at 30 °C was taken from Litovitz et al.12 Other investigators13,14 report very similar values. The elastic parameters for PMMA are from Penciu et al.2 The results for CdSe in water are presented in Figures 1-3. Rather than plotting ω, we use a dimensionless frequency ξ ) ωR/VLp for l ) 0 and η ) ωR/VTp for l > 0, where R is the radius of the nanoparticle. It is more appropriate to use ξ to express the frequency of (SPH,0) modes since their frequency depends primarily on VLp and less on VTp.15 By contrast, most

TABLE 2: Selected Vibrational Frequencies (in cm-1) of a Model of the Satellite Tobacco Mosaic Virus in Aira water µ)ζ)0 µ)0 ζ)0

(SPH,0)

(SPH,1)

(SPH,2)

(SPH,3)

3.666, 0.319 3.618 3.663, 0.318 3.620, 0.004

2.376, 0.177 2.200 2.223, 0.057 2.372, 0.235

1.450, 0.131 1.447 1.448, 0.004 1.449, 0.131

2.417, 0.261 2.385 2.386, 0.008 2.416, 0.263

a The damping originates from the viscous water core. For damped modes, the real and imaginary parts of the frequency are given. The rows of the table correspond to (1) water, (2) nonviscous water, (3) water with only bulk viscosity, and (4) water with only shear viscosity.

other modes have frequencies depending primarily on VTp, making η more natural to use. The values of ξ or η are complex numbers. To depict them, the value of the real part is shown as a solid line (red online). The value of the imaginary part is shown as a band that extends from Re(ξ) - Im(ξ) to Re(ξ) + Im(ξ) (or from Re(η) - Im(η) to Re(η) + Im(η)) on the plot. These bands are plotted with arbitrary filling patterns (and colors) in order to be able to see all the features even when they overlap. Without viscosity (and also, in the limit of large diameter), ξ and η do not vary with the nanoparticle diameter. Therefore, all of the functional dependence on diameter in all these figures is due to the viscosity of water (both shear and bulk). The applicability of this calculation is limited to frequencies for which the hydrodynamic description of water is valid.16 The viscosity may only be regarded as a frequency-independent quantity for wavevector q below 2 nm-1. Since ω ) cq and c ) 1498 m/s for water, the limiting frequency is approximately 3.0 × 1012 rad/s (or 16 cm-1). Above this, the viscous coefficients of water becomes q-dependent. In each figure, there is a cross-hatched black region in the top left corner corresponding to mode frequencies above this limit. The predictions of the model are not to be trusted in this region. Results for PMMA spheres in water are shown in Figures 4-6. As another application, we consider a simplified model of a satellite tobacco mosaic virus (STMV). This virus is generally icosahedral in shape. Approximating it as a sphere is plausible if only the few lowest lying vibrational modes are to be calculated. The central core of the virus is a water sphere, 4 nm in radius. Surrounding this is a 2 nm thick layer of DNA, whose density is 1.3 g/cm3, and whose sound speeds are 2800 and 1400 m/s. Finally, there is a 2.5 nm thick layer of protein with density 1.2 g/cm3 and sounds speeds 1800 and 915 m/s. The virus is surrounded by a vacuum. The free vibrations of this virus are damped as a result of the viscosity of the water core. The frequencies of some selected vibrational modes of this model virus are given in Table 2. For comparison, the mode frequencies are also given for the case where the viscosity is neglected. In this situation, the modes are undamped. In addition, we give the frequencies for the cases where either the bulk viscosity or shear viscosity is still included. The effect of the viscosity of the water core is to damp the vibrational modes, although not strongly. Furthermore, the origin of the damping depends on the nature of the mode. For the (SPH,0) mode, the damping is primarily due to the bulk viscosity of the water. For the other modes, the damping comes mostly from the shear viscosity. 5. Discussion Figure 1 shows how the viscosity of the water increases the damping of (SPH,0) modes of a CdSe nanoparticle when the

7460 J. Phys. Chem. B, Vol. 111, No. 25, 2007

Figure 1. Dimensionless frequency ξ ) ωR/VLp plotted versus nanoparticle diameter, 2R. The solid lines (red online) show Re(ξ). The bands surrounding the solid lines (alternately green, red, or blue online) extend from Re(ξ) - Im(ξ) to Re(ξ) + Im(ξ). The frequencies shown are for (SPH,l ) 0) modes of a CdSe nanoparticle in water. The black cross-hatched region in the top left is where mode frequency exceeds the limit for which these calculations are reliable. At the right edge of the figure, the vibrational frequencies are shown in the absence of viscosity, as bars going from Re(ξ) - Im(ξ) to Re(ξ) + Im(ξ).

Saviot et al.

Figure 3. As in Figure 1 but for (SPH,l ) 2) modes of a CdSe nanoparticle in water.

Figure 4. As in Figure 1 but for (SPH,l ) 0) modes of a PMMA nanoparticle in water.

Figure 2. As in Figure 1 except that dimensionless frequency η ) ωR/VTp is plotted versus nanoparticle diameter, 2R, for (SPH,l ) 1) modes of a CdSe nanoparticle in water.

diameter gets down to 10 nm or less. For example, for a 10 nm diameter nanoparticle, the damping of the (SPH,0,0) mode is approximately 50% greater once viscosity is taken into account. However, our approach cannot be trusted for the (SPH,0,0) mode when the diameter is below 6 nm, since the hydrodynamic description of water breaks down. It is necessary to take viscosity into account when analyzing experiments involving the excitation of (SPH,0) modes of a nanoparticle using femtosecond laser pulses. Figure 2 shows the effect of viscosity on (SPH,1) modes. The main new feature of (SPH,1) modes is that some of them have almost negligible damping in the absence of viscosity. However, some (SPH,1) modes are moderately damped in the absence of viscosity. The reason for the difference is because spheroidal modes can be approximately categorized as either “mostly longitudinal” or “mostly transverse”.15 For mostly transverse modes, there is very little radial motion of the surface. As a result, such modes have only slight loss of energy due to contact with the water. (SPH,1) modes have the correct symmetry to absorb terahertz electromagnetic radiation17 as has been observed in TiO2 nanopowders.18 The lowest branch with

very damped modes does not correspond to a free sphere model mode and is similar to the “matrix modes” previously seen for matrix embedded nanoparticles.19 Similar damped modes exist in the other figures. For a solid matrix, restoring forces transform the translation and rotation of the nanoparticle into “rattling” (SPH,1) and “libration” (TOR,1) modes respectively. Such modes do not exists for a nanoparticle in water where damped translations and rotations exist instead. Figure 3 is for (SPH,2) modes of CdSe in water. These modes are observable, in principle, using Raman scattering. However, the strong background scattering from the water makes this difficult. Figures 4-6 are for PMMA in water. These modes are relatively strongly damped even in the absence of viscosity as a result of water and PMMA having similar mass densities and longitudinal sound speeds. For large diameters, the medium surrounding the nanoparticle is softer and less dense than the nanoparticle, and the system behaves mostly as a free sphere. As the diameter decreases, the real and imaginary parts of the sound speeds increase (see eqs 16 and 17) which means that the water becomes, in effect, “harder” than the nanoparticle but also that damping in the water is more important. This is the reason why the free sphere model and calculations not taking into account viscosity are no longer good approximations. The primary effect of viscosity at small diameters is to shift the frequency down. The vibrational frequencies of a model STMV virus in air (as opposed to water) illustrates the different qualitative effects

Damping of Confined Acoustic Phonons

J. Phys. Chem. B, Vol. 111, No. 25, 2007 7461 In general, it is clear that viscosity has important effects on the vibrational modes of nanoparticles in the entire nanoscale size range. The most dramatic influence of viscosity is for vibrational modes for which surface motion is primarily transverse, since these modes cannot couple to sound waves in the water in the absence of viscosity. We have only presented results using the values of the viscosities at 30 °C and one atmosphere. However, there is a significant increase of both viscosity coefficients in water when temperature is lowered.14 Reducing the temperature to 10 °C roughly doubles both coefficients. The bulk viscosity rises dramatically as 0 °C is approached.

Figure 5. As in Figure 1 but for (SPH,l ) 1) modes of a PMMA nanoparticle in water.

Acknowledgment. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The authors want to acknowledge A. Mermet and T. M. Liu for very interesting discussions. References and Notes

Figure 6. As in Figure 1 but for (SPH,l ) 2) modes of a PMMA nanoparticle in water.

of shear and bulk visosity of the virus’s central liquid core. Bulk viscosity is the cause of energy loss in (SPH,0) modes. Shear viscosity dominates energy loss for other modes if they are primarily transverse. This approach could also be applied to nonspherical viruses such as the M13 bacteriophage. This virus is approximately cylindrical in shape, with diameter 6.5 nm and length 860 nm. The importance of viscosity in the description of water demonstrated here for spherical symmetry is certainly relevant also in the cylindrical symmetry for such small sizes.

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