5.1 Comparisons with the available theoretical and experimental results

Table 1.

Comparison of the quality factor $Q$ and vibration frequency $f=\unicode[STIX]{x1D714}_{r}/2\unicode[STIX]{x03C0}$ of the fundamental (or first-order) breathing mode with available theoretical predictions for isotropic gold nanospheres (40 nm and 10 nm radii) vibrating in non-viscous compressible, viscous compressible, linear Maxwell and compressional non-Newtonian fluids. The fluid media include pure water ( $\unicode[STIX]{x1D712}=0$ ) and glycerol–water mixture ( $\unicode[STIX]{x1D712}=0.56$ ); the reader is referred to SM-III for details of their material constants. $^{a}$  The exact theoretical predictions are from Galstyan et al. (Reference Galstyan, Pak and Stone2015).

The analysis based on the state-space formalism combined with the approximate laminate technique (hereafter abbreviated as state-space method or SSM) along with Muller’s method will be verified in this subsection in terms of accuracy by comparison with the available theoretical and experimental results. Employing the proposed method, we first calculate the vibration frequency $f=\unicode[STIX]{x1D714}_{r}/2\unicode[STIX]{x03C0}$ and the quality factor $Q$ of the fundamental (or first-order) breathing mode ( $n=0$ ) of isotropic gold nanospheres of two different radii (40 nm and 10 nm) vibrating in two glycerol–water mixtures. The numerical results are given in table 1 for non-viscous compressible, viscous compressible, linear Maxwell and compressional non-Newtonian fluid models. The theoretical predictions based on the conventional displacement method in Galstyan et al. (Reference Galstyan, Pak and Stone2015) are also presented in table 1 for comparison and validation purposes. As described in § 2, the linear Maxwell model without the compressional relaxation process was utilized by Galstyan et al. (Reference Galstyan, Pak and Stone2015) to study the breathing mode. Thus, the results predicted by the compressional non-Newtonian model are also shown in table 1 to highlight the compressional relaxation effect. It can be seen that the calculated results based on SSM agree exceptionally well with those by the exact displacement method. In addition, the compressional relaxation process will have a significant influence on the vibration quality factor, especially for a nanosphere with a smaller radius and a fluid medium with a larger viscoelasticity.

Table 2.

Comparison of the vibration frequency $f=\unicode[STIX]{x1D714}_{r}/2\unicode[STIX]{x03C0}$ for various experimentally excited and observed modes for different isotropic nanospheres vibrating in different surrounding media. The experimental results concerning 40 nm gold nanoparticles in water and 50 nm free influenza A virions, respectively, are taken from measurements conducted by Ruijgrok et al. (Reference Ruijgrok, Zijlstra, Tchebotareva and Orrit2012) and Liu et al. (Reference Liu, Chen, Wang, Wang, Luo, Chen, Liu and Sun2009). These experimental literature data are displayed in square brackets.

Combining ultrafast pump–probe spectroscopy with optical trapping, Ruijgrok et al. (Reference Ruijgrok, Zijlstra, Tchebotareva and Orrit2012) conducted an elaborate experiment to measure the dynamic responses of a single gold nanoparticle (40 nm radius) immersed in water and detected the fundamental and higher-order breathing modes and the first-order quadrupolar mode. Through the dipolar coupling to confined acoustic modes, the microwave resonant absorption (MRA) of virions was confirmed by Liu et al. (Reference Liu, Chen, Wang, Wang, Luo, Chen, Liu and Sun2009) and the absorption peaks of the first two dipolar modes were observed for influenza A virions (with 50 nm radius, effective longitudinal wave velocity $2450~\text{m}~\text{s}^{-1}$ and Poisson’s ratio 0.33). Hence, the theoretically predicted vibration frequency based on the compressional non-Newtonian fluid model can be compared with those experimental results, as shown in table 2. Since different modes of nanoparticles have been experimentally excited under proper external stimuli, the comparison in table 2 includes the fundamental or higher-order breathing ( $n=0$ ), dipolar ( $n=1$ ) and quadrupolar ( $n=2$ ) modes. It is found that the theoretical predictions based on the SSM agree quite well with the experimental results.

Thus, we can use the present SSM and Muller’s method to obtain accurate numerical results with a high precision. In the following calculations, the number of equally discretized thin layers is taken to be 50, for which the numerical results can be assumed to be highly accurate. If not otherwise stated, a gold nanosphere with radius $b=r_{N}$ immersed in different glycerol–water mixtures is considered; the associated material properties are all provided in SM-III for completeness. In addition, in what follows, we will only pay attention to the smallest non-zero root (i.e. the first-order radial mode) for each angular mode  $n$ and the higher-order ones can be readily obtained if needed.

5.2 Effects of fluid viscosity and compressibility

In this subsection, the vibration characteristics in vacuum, non-viscous incompressible, non-viscous compressible, viscous incompressible and viscous compressible fluids will be shown separately for comparison and illustration.

Figure 2.

Normalized vibration frequencies versus the angular mode number $n$ of an isotropic gold sphere in a glycerol–water mixture ( $\unicode[STIX]{x1D712}=0.56$ ) using the Newtonian model: (a) torsional modes for two different radii 4 mm and 40 nm; and (b) spheroidal modes for 40 nm.

The effects of fluid viscosity and compressibility on the normalized vibration frequencies $\unicode[STIX]{x1D6FA}_{r}/\unicode[STIX]{x03C0}$ and damping components $\unicode[STIX]{x1D6FA}_{i}/\unicode[STIX]{x03C0}$ are depicted in figures 2 and 3, respectively, for the two classes of vibration of an isotropic solid sphere submerged in a glycerol–water mixture modelled by a Newtonian fluid. Note that the vibration attenuation originates from two aspects: energy dissipation due to the viscosity of the fluid, and outgoing energy propagation because of fluid compressibility (Ding & Chen Reference Ding and Chen1998; Galstyan et al. Reference Galstyan, Pak and Stone2015). Since the first class of vibration is an equi-volumetric motion, the fluid compressibility does not affect the free vibration characteristics of the sphere. Therefore, figures 2(a) and 3(a) only show the effect of fluid viscosity. One can observe from figure 2 that, for the first class of vibration, the vibration frequency for the purely torsional or rotary mode ( $n=1$ ) is higher than that of the mode $n=2$ , and the frequency of the torsional mode increases with the increasing angular mode number when $n\geqslant 2$ ; the vibration frequencies of the second class corresponding to the non-breathing modes ( $n=1{-}6$ ) are lower than that of the breathing mode $n=0$ . As mentioned before, different vibration modes of nanoparticles have been excited and observed in experiments (Liu et al. Reference Liu, Chen, Wang, Wang, Luo, Chen, Liu and Sun2009; Crut et al. Reference Crut, Maioli, Del Fatti and Vallée2015), and hence it is extremely important to theoretically study the vibration characteristics of general vibration modes other than the simple breathing mode.

Figure 3.

Normalized damping components versus the angular mode number $n$ of an isotropic gold sphere in glycerol–water mixture ( $\unicode[STIX]{x1D712}=0.56$ ) using the Newtonian model: (a) torsional modes for 4 mm and 40 nm; and (b,c) spheroidal modes for (b) 4 mm and (c) 40 nm.

The results in figures 2 and 3 show that fluid compressibility and viscosity have little effect on the resonant frequency of the two classes of vibration, but play a more obvious role in the damping component. This indicates that the free vibration frequency is hardly affected by the properties of the surrounding fluid while the attenuation behaviour is strongly dependent on the fluid compressibility and viscosity. Specifically, for the first class of vibration, the damping component increases as the mode number increases and the radius of the sphere decreases, which means that the effect of viscosity on the attenuation of higher-order angular modes and smaller nano-sized spheres is more significant than that of lower-order angular modes and larger spheres (say, at a macroscale). For the second class of vibration, it can be seen from figure 3(b) that, for a macroscopic dimension ( $b=4~\text{mm}$ ), the effect of viscosity is small and the attenuation is mainly fixed by the fluid compressibility. The attenuation of the higher-order modes is smaller than that of the breathing mode. For the nanosphere ( $b=40~\text{nm}$ ) shown in figure 3(c), the result corresponding to the non-viscous compressible case is hardly changed compared with that of a macroscopic size in figure 3(b). Nonetheless, the effect of viscosity becomes more pronounced, eventually leading to an increase in the attenuation of the second class of vibration over the entire mode range of interest.

Figure 4.

Quality factors of the second class of vibration versus the logarithmic quasi-Reynolds number $\log _{10}\tilde{r}$ of an isotropic gold sphere in glycerol–water mixture ( $\unicode[STIX]{x1D712}=0.56$ ) using the Newtonian model for the breathing $n=0$  (a), dipolar $n=1$  (b) and quadrupolar $n=2$  (c) modes. The quasi-Reynolds number is defined as $\tilde{r}=\unicode[STIX]{x1D70C}_{f}c_{f}b/\unicode[STIX]{x1D702}$ .

In order to clearly show the dependence of vibration characteristics on the size of the sphere and the fluid viscosity, we display the variations of the quality factor of the submerged isotropic gold sphere with the logarithmic quasi-Reynolds number $\log _{10}\tilde{r}$ in figure 4 for the breathing ( $n=0$ ), dipolar ( $n=1$ ) and quadrupolar modes ( $n=2$ ) of the second class. It is interesting to note that, for a non-viscous compressible fluid, the damping of which is solely due to the propagation of acoustic waves away from the vibrating sphere, the quality factor does not change with the quasi-Reynolds number $\tilde{r}$ , which means that there is no size effect. Nevertheless, when the viscosity effect is considered, the vibration characteristics vary with $\tilde{r}$ , exhibiting the size effect. In fact, in addition to the radius $b$ of the solid sphere, there exists another intrinsic size $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70C}_{f}c_{f}$ of the fluid due to the presence of fluid viscosity. This is the right reason for the appearance of the size-dependent characteristics. Specifically, a general variation trend that the quality factors of all vibration modes of the sphere in the viscous fluid go down with the decrease of $\tilde{r}$ is observed in figure 4. Since the density and sound speed do not alter significantly for different fluids, the reduction of the quasi-Reynolds number $\tilde{r}=\unicode[STIX]{x1D70C}_{f}c_{f}b/\unicode[STIX]{x1D702}$ can be classified into two cases, namely, the decrease of the radius $b$ of the sphere, and the increase of the fluid viscosity $\unicode[STIX]{x1D702}$ . As a result, when $\tilde{r}$ is infinitely large, the attenuation induced by fluid viscosity becomes small, the attenuation behaviour mainly relies on fluid compressibility, and the vibration quality factor for the viscous compressible fluid approaches the non-viscous compressible result. However, when $\tilde{r}$ reduces gradually, the viscosity-induced attenuation has a dramatic increase. In this case, the damping component is mainly determined by the viscosity, and thus the quality factor asymptotically tends towards the viscous incompressible result (see figure 4), which has a limiting value of 0.5. In short, when the size of the vibrating sphere decreases to the nanometre scale or the shear viscosity of the fluid increases, fluid viscosity has a significant influence on the vibration characteristics of the FSI system and must be taken into account (Galstyan et al. Reference Galstyan, Pak and Stone2015).

For the first class of vibration, numerical calculations have also been conducted for a submerged gold sphere for different torsional modes and glycerol–water mixtures. The results are qualitatively similar to those for the second class of vibration (see figure S2 in supplementary material SM-V).

Figure 5.

Normalized vibration frequency (a) and logarithmic quality factor (b) of the breathing mode ( $n=0$ ) versus the logarithmic quasi-Reynolds number $\log _{10}\tilde{r}$ for an isotropic sphere immersed in a compressible viscoelastic fluid with different quasi-Deborah numbers $De_{sh}=0$ (compressible Newtonian model), 1, 10, $10^{2}$ and $10^{3}$ . Results are provided based on the linear Maxwell model (dashed lines) and compressional non-Newtonian model (solid lines) for the dimensionless quantities $\tilde{\unicode[STIX]{x1D70C}}=0.05$ , $\tilde{v}=0.8$ , $\unicode[STIX]{x1D6FE}=1$ and Poisson’s ratio $\unicode[STIX]{x1D708}=0.42$ . The quasi-Deborah numbers are chosen as $De_{comp}=De_{sh}$ for the compressional non-Newtonian model.

5.3 Effect of fluid viscoelasticity

Now we turn to elucidating the effect of fluid viscoelasticity on the vibration characteristics of the submerged nanosphere. When viscoelasticity is present, the Deborah number $De=\unicode[STIX]{x1D706}\unicode[STIX]{x1D714}$ is usually adopted (Chakraborty & Sader Reference Chakraborty and Sader2015; Chakraborty et al. Reference Chakraborty, Hartland, Pelton and Sader2018), where $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D714}$ represent the characteristic molecular relaxation time and vibration circular frequency of the structure. Nonetheless, since $\unicode[STIX]{x1D714}$ (especially the damping component) strongly depends on the viscoelastic properties of the surrounding fluid, we will scale the shear and compressional relaxation times $\unicode[STIX]{x1D706}_{sh}$ and $\unicode[STIX]{x1D706}_{comp}$ by the circular frequency $\unicode[STIX]{x1D714}_{vac}$ of the structure in vacuum and thus define two quasi-Deborah numbers $De_{sh}=\unicode[STIX]{x1D706}_{sh}\unicode[STIX]{x1D714}_{vac}$ and $De_{comp}=\unicode[STIX]{x1D706}_{comp}\unicode[STIX]{x1D714}_{vac}$ . Note that $\unicode[STIX]{x1D714}_{vac}$ is determined only by the structure itself and is not affected by the fluid.

Based on the linear Maxwell ( $De_{comp}=0$ ) and compressional non-Newtonian ( $De_{comp}\neq 0$ ) models, the variations with the logarithmic quasi-Reynolds number $\log _{10}\tilde{r}$ of the normalized vibration frequency $\unicode[STIX]{x1D714}_{r}/\unicode[STIX]{x1D714}_{vac}$ and the logarithmic quality factor $\log _{10}Q$ of a submerged sphere are depicted in figure 5 for the breathing mode and different quasi-Deborah numbers. When the compressional non-Newtonian model is employed, the condition $De=De_{sh}=De_{comp}$ is taken as a good approximation for some real fluids (Chakraborty & Sader Reference Chakraborty and Sader2015; Chakraborty et al. Reference Chakraborty, Hartland, Pelton and Sader2018). It can be seen from figure 5 that both the vibration frequency and quality factor predicted by these two models for all quasi-Deborah numbers asymptotically tend towards the non-viscous compressible results for large $\tilde{r}$ (small viscosity effect) and approach the incompressible viscoelastic results for small $\tilde{r}$ due to the viscoelasticity-dominated influence (Chakraborty & Sader Reference Chakraborty and Sader2015). The quality factor corresponding to the Newtonian model ( $De=0$ ) shown in figure 5(b) declines monotonically with decreasing $\tilde{r}$ , which is similar to the result in figure 4(a). However, the vibration frequency for $De=0$ in figure 5(a) exhibits a non-monotonic behaviour as a function of $\tilde{r}$ , which may be attributed to the complex mutual interaction of fluid viscosity, compressibility and structural size (Chakraborty & Sader Reference Chakraborty and Sader2015).

Furthermore, results in figure 5 display that the predictions based on the linear Maxwell and compressional non-Newtonian models for finite quasi-Deborah numbers (i.e. $De\neq 0$ ) demonstrate a dramatic difference from the Newtonian results. Specifically, the vibration frequency increases remarkably with a reduction in $\tilde{r}$ , especially in the low- $\tilde{r}$ region. Additionally, instead of a reduction in quality factor with decreasing $\tilde{r}$ , the quality factor has a monotonically reverse increase after a critical value $\tilde{r}_{c}$ is reached. These seemingly unintuitive behaviours originate from the significant effect of fluid elasticity: the stored energy in the FSI system overly compensates the energy dissipation induced by fluid viscosity and compressibility, which results in the remarkable rise in the vibration frequency and quality factor in figure 5 and will be discussed further below for a real FSI system (i.e. a gold nanosphere vibrating in a glycerol–water mixture).

Moreover, it is obvious from figure 5 that for a fixed $\tilde{r}$ of a moderate or small magnitude, the vibration frequency provided by the compressional non-Newtonian model is larger than that predicted by the linear Maxwell model, while the quality factor shows an opposite trend. The reason for this phenomenon can be explained as follows. For the breathing mode with the nature of compressional motion, the compressional elasticity stiffens the FSI system, leading to the increase of vibration frequency. Nonetheless, the compressional relaxation effect enhances the compressibility effect and thus increases the energy radiation (the calculated damping component of the circular frequency by the compressional non-Newtonian model is larger than that based on the linear Maxwell model), producing a decrease in the quality factor of the breathing mode, especially for high quasi-Deborah numbers. This phenomenon will also be investigated further below for a specific FSI system by means of the spatial distribution of the stored energy density.

Figure 6.

Variations of the quality factor with the angular mode number $n$ of a 40 nm isotropic gold nanosphere for different glycerol–water mixtures using both Newtonian and linear Maxwell models: (a) spheroidal modes; and (b) torsional modes.

To highlight the difference between the Newtonian and non-Newtonian fluid models, figure 6 shows the variation of the quality factor of two classes of vibration with the angular mode number for a 40 nm isotropic gold nanosphere in different glycerol–water mixtures. It can be seen that, for the two classes of vibration and low glycerol mass fraction $\unicode[STIX]{x1D712}$ (i.e. the effect of viscoelasticity is small), the predictions based on the two models are not very different when the mode is lower than a certain critical mode. For instance, the critical modes for the second class of vibration corresponding to $\unicode[STIX]{x1D712}=0$ and $\unicode[STIX]{x1D712}=0.56$ are $n=6$ and $n=3$ , as shown in figure 6(a). Beyond the critical mode, the difference between these two models increases gradually with increasing angular mode number. It is worth pointing out that the critical mode number varies and depends on the size and glycerol mass fraction, which will be discussed in detail below. Furthermore, for high glycerol mass fractions such as $\unicode[STIX]{x1D712}=0.8$ and $\unicode[STIX]{x1D712}=0.85$ , the obtained results based on the two models are extremely different over the entire mode range, especially for the higher-order mode numbers. Specifically, as the mode number increases, the quality factor predicted by the Newtonian model monotonically decreases to a limiting value of 0.5, while that based on the linear Maxwell model first decreases and then increases reversely.

The variation trend predicted by the compressional non-Newtonian model accounting for both the shear and compressional relaxation effects is qualitatively consistent with that of the linear Maxwell model, and hence it is not shown here for brevity. Instead, we will only demonstrate the difference of the predictions based on the two fluid models. Note that, for the first class of vibration, the compressional relaxation effect does not alter the free vibration characteristics of the submerged sphere due to its equi-volumetric nature. However, for the second class of vibration, the compressional relaxation process indeed affects the vibration frequency and quality factor. The difference of the quality factor of the second class of vibration predicted by these two fluid models is plotted in figure 7 for different glycerol mass fractions. It is found that, for low mass fractions, the compressional relaxation still has a noticeable influence on the breathing mode $n=0$ because of its compressional motion behaviour, but its effect on the higher-order modes is less significant compared with the breathing mode and is negligible. For high mass fractions, the vibration characteristics strongly depend upon the compressional relaxation process over the whole mode range.

Figure 7.

Variations of the quality factor difference $\unicode[STIX]{x0394}Q=Q_{1}-Q_{2}$ with the angular mode number $n$ of a 40 nm isotropic gold nanosphere for different glycerol–water mixtures using two different non-Newtonian models, where $Q_{1}$ is the quality factor predicted based on the compressional non-Newtonian model, while $Q_{2}$ is that by the linear Maxwell model.

Figure 8.

Spatial distributions of the normalized exceptional maximum stored energy density $E^{exc}$ in a vibrating isotropic gold nanosphere ( $b=40~\text{nm}$ ) and in the surrounding glycerol–water mixture for the breathing mode $n=0$ using the linear Maxwell model (a,c) and the compressional non-Newtonian model (b,d): (a,b) $\unicode[STIX]{x1D712}=0$ (pure water); and (c,d) $\unicode[STIX]{x1D712}=0.8$ . The colour scale represents the exceptional maximum stored energy density normalized by the energy density at the outer surface of the nanosphere. The radial thickness of the fluid computational domain is limited within  $2b$ .

Moreover, for different glycerol–water mixtures, figure 8 depicts the spatial distributions of the normalized exceptional maximum stored energy density $E^{exc}$ (see SM-IV for more details) in the vibrating gold nanosphere as well as in the surrounding fluid for the breathing mode $n=0$ predicted by the linear Maxwell model and the compressional non-Newtonian model. Since the breathing mode has only the radial displacement component (which is independent of $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D711}$ ), the spatial distribution of the energy density only varies along the $r$ -direction and does not change along the $\unicode[STIX]{x1D703}$ -direction, as shown in figure 8. One can observe that the compressional relaxation effect on the energy density distribution of the breathing mode is more significant for a high glycerol mass fraction ( $\unicode[STIX]{x1D712}=0.8$ ) than a low mass fraction ( $\unicode[STIX]{x1D712}=0$ ). In particular, since the breathing mode is a spherically symmetric compressional motion, the compressional relaxation enhances the outward energy propagation due to the vibrating nanosphere and increases the energy attenuation, leading to a decrease in the quality factor of the breathing mode (see figure 7). Overall, for complex fluids with considerable viscoelastic effect and for the compression-dominated flow induced by compressional motion, it is necessary to use the compressional non-Newtonian model instead of the Newtonian and linear Maxwell models to accurately predict the vibration characteristics.

Now the compressional non-Newtonian model is utilized to explore under what conditions the viscoelastic effect should be taken into consideration. For different vibration modes and different radii of the nanosphere, figure 9 displays the quality factors $Q$ predicted by both the Newtonian and compressional non-Newtonian models as a function of the glycerol mass fraction $\unicode[STIX]{x1D712}$ for the second class of vibration. It is found that the quality factor based on the Newtonian model monotonically decreases to the limiting value of 0.5 with the increase of $\unicode[STIX]{x1D712}$ . However, as the mass fraction increases to a certain critical value $\unicode[STIX]{x1D712}_{c}$ , the quality factor given by the compressional non-Newtonian model no longer decreases, but remains almost unchanged (e.g. the breathing mode $n=0$ ), and even goes up reversely for some modes (e.g. the quadrupolar mode $n=2$ and the mode $n=6$ ). In fact, as Pelton et al. (Reference Pelton, Chakraborty, Malachosky, Guyot-Sionnest and Sader2013) pointed out, for low mass fractions with weak fluid elastic effect, the dissipation due to viscosity and compressibility results in the decrease of the quality factor and the results for both models agree with each other quite well. For mass fractions higher than the critical value $\unicode[STIX]{x1D712}_{c}$ , the stored energy in the fluid can compensate or even exceed the dissipated energy induced by viscosity and compressibility. Therefore, the quality factor no longer declines and can even increase with increasing glycerol mass fraction, which was also observed by Galstyan et al. (Reference Galstyan, Pak and Stone2015) for the breathing mode of a gold nanosphere vibrating in a linear Maxwell fluid. Essentially the same analysis can be conducted for torsional modes of the first class of vibration (see figure S3 in SM-V).

Figure 9.

Quality factor $Q$ of the second class of vibration as a function of the glycerol mass fraction $\unicode[STIX]{x1D712}$ for the isotropic gold nanosphere with different radii $b=40~\text{nm}$ (a–c) and $b=20~\text{nm}$ (d–f) and different spheroidal modes $n=0$ (a,d), $n=2$  (b,e) and $n=6$ (c,f) using both the Newtonian and compressional non-Newtonian models.

Figure 10.

Spatial distributions of the normalized exceptional maximum stored energy density $E^{exc}$ of the second class of vibration in a 40 nm isotropic gold nanosphere and in the surrounding glycerol–water mixture for the breathing mode $n=0$ (a–c) and quadrupolar mode $n=2$ (d–f) using the compressional non-Newtonian model: (a,d)  $\unicode[STIX]{x1D712}=0$ (pure water); (b,e) $\unicode[STIX]{x1D712}=0.56$ ; and (c,f) $\unicode[STIX]{x1D712}=0.85$ . The colour scale represents the exceptional maximum stored energy density normalized by the energy density at the north pole of the nanosphere for each mode and mass fraction. The radial thickness of the fluid computational domain is limited within  $2b$ .

Figure 11.

Quality factor $Q$ of the second class of vibration as a function of the logarithmic radius $\log _{10}[b\,(\text{nm})]$ for an isotropic gold nanosphere in different glycerol–water mixtures using both the Newtonian and compressional non-Newtonian models for different spheroidal modes: (a) breathing mode $n=0$ ; (b) dipolar mode $n=1$ ; (c) quadrupolar mode $n=2$ ; and (d)  $n=1$ (enlarged).

In addition, one can observe from figure 9 that the critical glycerol mass fraction $\unicode[STIX]{x1D712}_{c}$ depends on the mode number and the radius of the nanosphere. In fact, equation (2.4) demonstrates that the elastic effect of the fluid will become more obvious if $|\unicode[STIX]{x1D714}\unicode[STIX]{x1D706}_{sh}|\geqslant 1$ or $|\unicode[STIX]{x1D714}\unicode[STIX]{x1D706}_{comp}|\geqslant 1$ , i.e. when the relaxation time scale of the fluid $\unicode[STIX]{x1D706}_{sh}$ or $\unicode[STIX]{x1D706}_{comp}$ is comparable to the vibration time scale $1/\unicode[STIX]{x1D714}_{r}$ (Galstyan et al. Reference Galstyan, Pak and Stone2015). Based on the given vibration frequency $\unicode[STIX]{x1D714}_{r}$ , one can therefore estimate the critical glycerol mass fraction $\unicode[STIX]{x1D712}_{c}$ beyond which the prediction by the compressional non-Newtonian model will deviate considerably from that by the Newtonian model. For example, for the second class of vibration of a 40 nm gold nanosphere, the vibration frequencies $f=\unicode[STIX]{x1D714}_{r}/2\unicode[STIX]{x03C0}$ corresponding to the spheroidal modes $n=0$ , 2 and 6 in vacuum are calculated as $f\approx 38~\text{GHz}$ , 13 GHz and 35 GHz, respectively. Note that it can be numerically confirmed that the vibration frequency is not significantly affected by the surrounding fluid considered here. The critical relaxation times determined from the condition $|\unicode[STIX]{x1D714}\unicode[STIX]{x1D706}_{c}|=1$ are approximately $\unicode[STIX]{x1D706}_{c}\approx 4.2~\text{ps}$ , 12.2 ps and 4.5 ps, which correspond to the critical mass fractions $\unicode[STIX]{x1D712}_{c}$ in the range of 0.5–0.7 (referring to table S1 in the supplementary material for the shear and compressional relaxation times of different glycerol–water mixtures). These are consistent with the results in figure 9. We can also execute similar analysis for the torsional modes of the first class of vibration (see figure S3 in SM-V). Interestingly, for the spheroidal mode $n=6$ with $b=20~\text{nm}$ (see figure 9 c) and for the torsional mode $n=1$ with $b=10~\text{nm}$ (see figure S3(c) in SM-V), the deviation of the predictions based on the two models shows up even when the glycerol mass fraction is close to zero (i.e. pure water), indicating that, for higher-order angular modes or smaller spherical radii with high vibration frequencies, the viscoelastic effect on fluid–solid coupled vibrations becomes significant even in a low-viscosity fluid. As a consequence, the FSI vibration characteristics at nanoscale can be exploited to measure the viscoelastic properties of low-viscosity fluids (such as pure water) with faster relaxation times.

In order to evidently demonstrate the stored energy evolution with the glycerol mass fraction, the spatial distributions of $E^{exc}$ in a 40 nm isotropic nanosphere and different surrounding mixtures are illustrated in figure 10 for the breathing ( $n=0$ ) and quadrupolar ( $n=2$ ) modes of the second class of vibration using the compressional non-Newtonian model. Note that for axisymmetric modes (i.e. $m=0$ ), it is only necessary to give the energy distribution in the $r{-}\unicode[STIX]{x1D703}$ plane. For the breathing mode, the energy distribution does not change along the $\unicode[STIX]{x1D703}$ -direction, while for the quadrupolar mode, the displacement and stress components depend on $\unicode[STIX]{x1D703}$ , resulting in the variation of the energy distribution with $\unicode[STIX]{x1D703}$ . However, the energy distribution of the quadrupolar mode is still symmetric, as shown in figure 10(d–f). It can be seen that, as expected, more energy is stored in the surrounding fluid due to the increasing elastic effect of the fluid as the glycerol mass fraction increases, which can compensate or even exceed the dissipated energy. The overall stiffness of the FSI system also increases with increasing glycerol mass fraction, which ultimately leads to the increase in the vibration frequency (not shown here for simplicity).

For different glycerol mass fractions and different vibration modes, the variations of the quality factor of the second classes of vibration with the logarithmic spherical radius $\log _{10}[b\,(\text{nm})]$ are depicted in figure 11 using the Newtonian and compressional non-Newtonian fluid models. As mentioned above, the quality factor predicted by the Newtonian model monotonically decreases with decreasing radius in all cases because of the increasing viscous dissipation effect. In contrast, as the radius decreases from macroscale to nanoscale, the quality factor based on the compressional non-Newtonian model first decreases to a minimum, and then remains almost unchanged or even has an opposite rise after the radius decreases to a certain critical value $b_{c}$ . When the radius is larger than $b_{c}$ , the predictions of the two models are almost the same, since the vibration frequency is not high enough for the elastic effect of the fluid to play a role. For a macroscopic radius, the vibration characteristics approach the non-viscous compressible results. On the other hand, if the radius is smaller than $b_{c}$ , the extremely high vibration frequency enhances the elastic effect of the glycerol–water mixture, and hence the quality factor no longer decreases and can even increase with decreasing radius (see figure 11).

The critical radius $b_{c}$ depends on the glycerol mass fraction as well as the mode number, and it can be estimated using the relation $|\unicode[STIX]{x1D714}_{c}\unicode[STIX]{x1D706}|=1$ as described earlier. For instance, for the breathing mode in figure 11(a), the critical vibration frequencies $f_{c}=\unicode[STIX]{x1D714}_{rc}/2\unicode[STIX]{x03C0}$ corresponding to glycerol mass fractions $\unicode[STIX]{x1D712}=0.8$ , 0.56 and 0.36 can be calculated as $f_{c}\approx 5.8~\text{GHz}$ , 43 GHz and 85 GHz, respectively, according to the relaxation times listed in table S1 in SM. Utilizing the numerical results of the vibration frequency of gold nanospheres in vacuum for the breathing mode, the corresponding critical radii of the sphere are $b_{c}\approx 260~\text{nm}$ , 29 nm and 17 nm, which occur when $\log _{10}[b_{c}\,(\text{nm})]\approx 2.41$ , 1.46 and 1.23, respectively, as displayed in figure 11(a). The discussions can be similarly applied to the dipolar and quadrupolar modes as well as the torsional modes of the first class of vibration (see figure S4 in SM-V). To clearly demonstrate the size effect of the spherical particle on energy distributions, figure S5 in SM-V further shows the spatial distributions of $E^{exc}$ in the vibrating gold nanosphere with different radii and in the surrounding fluid for the breathing mode. Qualitatively similar physical phenomena to those in figure 11 can be observed. Thus, we can exploit this behaviour to properly select the size and vibration mode of nanostructures to measure the viscoelastic properties of a fluid with a given mass fraction.

5.4 Effect of solid surface layer

In general, single-crystal (or monocrystalline) nanoparticles are not isotropic and exhibit various degree of anisotropy in material properties. To save space in the paper, we demonstrate in SM-V-C the capability of the previously derived frequency equations (4.15)–(4.18) in studying the anisotropy effect of the nanosphere on the dynamic properties of the FSI system. The interested reader is therefore referred to SM-V-C for more details.

At nanoscale, it is well known that the atomic structure in or near the surface of a medium usually experiences a different local environment from that in the bulk counterpart, such that the material properties of the surface are distinct from those of the bulk (Cammarata Reference Cammarata1997). In addition, a nanoscale surface adhesion layer with a certain thickness can be artificially utilized to actively control the vibration behaviour of the nanostructures (Chang et al. Reference Chang, Wen, Chakraborty, Su, Zhang, Shuang, Nordlander, Sader, Halas and Link2015). Therefore, we consider here a concentric core–shell structure with different material properties in the core and shell. This core–shell model has been used to investigate the surface effect on the dynamic properties of nanostructures in vacuum (Chen et al. Reference Chen, Wu, Zhang and Zhang2014; Wu et al. Reference Wu, Zhang, Chen and Zhang2015, Reference Wu, Chen and Zhang2018). Now our interest is to employ the developed multilayered spherical model to examine how the surface effect of the nanosphere influences the vibration characteristics of the FSI system. The core–shell nanosphere is assumed to be cubic and has a spherical solid core of radius $b$ as well as a thin surface layer of thickness $h$ , which can be approximately taken as 1 nm in our calculations. Two dimensionless elastic and density parameters ( $r_{c}=c_{ij}^{(2)}/c_{ij}^{(1)}$ and $r_{\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D70C}_{s}^{(2)}/\unicode[STIX]{x1D70C}_{s}^{(1)}$ ) are adopted to study the surface effect for different surface material properties $c_{ij}^{(2)}$ and $\unicode[STIX]{x1D70C}_{s}^{(2)}$ , wherein $c_{ij}^{(1)}$ and $\unicode[STIX]{x1D70C}_{s}^{(1)}$ denote the material properties of the spherical core.

For the breathing mode of a 20 nm cubic gold nanosphere ( $r_{\unicode[STIX]{x1D70C}}=1$ ), the variations of the normalized vibration frequency, damping component and quality factor with the glycerol mass fraction are illustrated in figure 12 for different surface elastic parameters using the compressional non-Newtonian fluid model. It can be seen that the variation trend with the surface effect ( $r_{c}\neq 1$ ) is qualitatively similar to that without a surface effect ( $r_{c}=1$ ). With the increase of the surface elastic parameter, the vibration frequency of the breathing mode increases over the entire glycerol mass fraction range, but the damping component has an opposite variation trend, thus resulting in the rise of the quality factor. In fact, an increase in surface elasticity enhances the overall stiffness of the FSI system, causes more energy to be stored in the nanosphere, and hence raises the vibration frequency and quality factor. We also conduct the numerical analysis on the effect of surface density parameter on the FSI vibration characteristics (see SM-V-D for more details). The increasing surface mass density greatly weakens the effect of the surrounding fluid on the damping component and hence enhances the vibration quality factor, which is physically quite understandable.

Figure 12.

Normalized vibration frequency (a), damping component (b) and quality factor (c) of the breathing mode $n=0$ as functions of the glycerol mass fraction for a 20 nm cubic gold nanosphere with surface effect ( $r_{\unicode[STIX]{x1D70C}}=1$ and different surface elastic parameters $r_{c}$ ) using the compressional non-Newtonian model.

Now let us turn our attention to the surface effect on the variation trend of the vibration characteristics with the size of the nanosphere. For the breathing mode $n=0$ of a cubic gold nanosphere in a glycerol–water mixture ( $\unicode[STIX]{x1D712}=0.56$ ), the variations of the normalized vibration frequency, damping component and quality factor with the logarithmic radius $\log _{10}[b\,(\text{nm})]$ are illustrated in figure 13 using the compressional non-Newtonian model for different surface elastic and mass density parameters. It can be seen that, when there is no surface effect, the vibration characteristics tends towards the non-viscous compressible results for an infinitely large radius; with the radius decreasing, the vibration frequency hardly changes as described earlier, while the damping component gradually increases to a constant value because of the increasing viscoelastic effect, thus causing the quality factor to first decrease and then approach a stable value. When a surface effect is involved, the vibration characteristics for a macroscopic size tend towards the results without a surface effect, which is expected since the surface effect becomes negligible for a macroscopic sphere. However, when the radius of the sphere reduces to a critical value, the vibration characteristics begin to deviate from the results without a surface effect. For instance, the critical radii for quality factors in figures 13(c) and 13(f) are approximately equal to 32 nm and 56 nm, respectively.

Figure 13.

Normalized vibration frequencies (a,d), damping components (b,e) and quality factors (c,f) of the breathing mode $n=0$ of a cubic gold nanosphere with surface effect in a glycerol–water mixture ( $\unicode[STIX]{x1D712}=0.56$ ) as functions of the logarithmic radius using the compressional non-Newtonian model: (a–c) $r_{\unicode[STIX]{x1D70C}}=1$ and different surface elastic parameters; and (d–f) $r_{c}=1$ and different surface density parameters.

Furthermore, the results in figure 13(a–c) show that, if the surface layer becomes softer ( $r_{c}=0.5$ ), the vibration frequency decreases but the damping component further increases due to the increasing viscous dissipation, leading to a continuous reduction of the quality factor with decreasing radius. Nonetheless, for a stiffer surface layer ( $r_{c}=2$ ), as the radius decreases, the vibration frequency continuously increases while the damping component first increases and has an opposite decrease after reaching a maximum, which makes the quality factor go up remarkably after the radius reaches the critical value. In fact, the stiffer surface layer causes more elastic energy to be stored in the nanosphere, which exceeds the dissipated energy induced by fluid viscosity and compressibility, giving rise to the obvious reverse increase of the quality factor. Again, the heavier surface layer reduces the effect of fluid inertia on the vibration characteristics of the nanosphere, resulting in a significant rise of quality factor (see figure 13 f).

Therefore, when the surface layer is properly chosen in terms of stiffness and density, the quality factor is prone to achieving an opposite increase with decreasing nanosphere size for the breathing mode. To conclude, we can actively modulate the vibration characteristics of the nanosphere submerged in a viscoelastic fluid by surface engineering. Additionally, we can also exploit the surface-dependent nanoscale FSI vibrations to design new molecular and atomic sensors with high sensitivity (Pelton et al. Reference Pelton, Chakraborty, Malachosky, Guyot-Sionnest and Sader2013).

5.5 Effect of solid intrinsic damping

It is worth pointing out that the aforementioned theoretical and numerical investigations only take account of the radiation and viscous damping resulting from fluids, and assume that the nanospheres are purely linear elastic without damping. In practice, different nanoparticles used in experiments may be equipped with some intrinsic damping mechanisms, such as internal defects and dissipation in the capping layer surrounding the nanoparticle. This could be the reason leading to the discrepancy between the experimentally measured and theoretically predicted vibration quality factors (Pelton et al. Reference Pelton, Wang, Gosztola and Sader2011; Ruijgrok et al. Reference Ruijgrok, Zijlstra, Tchebotareva and Orrit2012; Chakraborty et al. Reference Chakraborty, van Leeuwen, Pelton and Sader2013).

When solid intrinsic damping is taken into account, the elastic tensor is no longer real but also has a complex part (i.e. the real elastic stiffness plus an imaginary damping stiffness). In this work, we will employ the hysteretic viscoelastic model (Quintanilla et al. Reference Quintanilla, Fan, Lowe and Craster2015). Specifically, the complex Young’s modulus $E_{Y}$ can be written as $E_{Y}=E_{r}-\text{i}E_{i}$ , where $E_{r}$ specifies the storage elastic modulus while $E_{i}$ indicates the loss modulus due to the damping. Here, $E_{i}$ is assumed to be frequency-independent. To consider frequency-dependent loss modulus, other linear viscoelastic models such as the Kelvin–Voigt model may be adopted (Quintanilla et al. Reference Quintanilla, Fan, Lowe and Craster2015). The quality factor only including the contribution from fluid dissipation and taking no account of solid intrinsic damping is referred to as fluid quality factor $Q_{fluid}$ , while $Q_{intrinsic}$ indicates the quality factor of nanoparticles with intrinsic damping in vacuum. The total quality factor $Q_{total}$ accounts for all damping mechanisms in both fluids and solids.

Figure 14.

(a) Intrinsic quality factor $Q_{intrinsic}$ and total quality factor $Q_{total}$ of the breathing mode as functions of the loss-to-storage modulus ratio $E_{i}/E_{r}$ for a 40 nm isotropic gold nanosphere vibrating in vacuum and in different glycerol–water mixtures. (b) Total quality factor $Q_{total}$ (solid lines) for $E_{i}/E_{r}=0.025$ and fluid quality factor $Q_{fluid}$ (dashed lines) without solid intrinsic damping as functions of the logarithmic radius $\log _{10}[b\,(\text{nm})]$ for the breathing mode of isotropic gold nanospheres in different glycerol–water mixtures. In (a,b) the glycerol–water mixtures are described by the compressional non-Newtonian model, and the experimentally measured $Q_{intrinsic}$ (blue square) and $Q_{total}$ (pink circle) values conducted by Ruijgrok et al. (Reference Ruijgrok, Zijlstra, Tchebotareva and Orrit2012) are also displayed for the case of water ( $\unicode[STIX]{x1D712}=0$ ). In (b) the short-dashed lines represent the non-viscous compressible results.

First of all, for the breathing mode of a 40 nm isotropic gold nanosphere vibrating in vacuum and glycerol–water mixtures, figure 14(a) shows the variations of $Q_{intrinsic}$ and $Q_{total}$ with the loss-to-storage modulus ratio $E_{i}/E_{r}$ . As expected, both $Q_{intrinsic}$ and $Q_{total}$ decline monotonically with increasing $E_{i}/E_{r}$ owing to the enhancement of solid intrinsic damping. Note that Ruijgrok et al. (Reference Ruijgrok, Zijlstra, Tchebotareva and Orrit2012) carried out the experimental measurement of damping for the fundamental breathing mode of 40 nm gold nanospheres optically trapped in water ( $\unicode[STIX]{x1D712}=0$ ). The measured average intrinsic and total quality factors are $Q_{intrinsic}=40\pm 4$ and $Q_{total}=22\pm 1.5$ , respectively, both highlighted in figure 14(a). It is obvious that the theoretically predicted intrinsic and total quality factors agree exceptionally well with those measured experimentally (Ruijgrok et al. Reference Ruijgrok, Zijlstra, Tchebotareva and Orrit2012). In addition, the loss-to-storage modulus ratio for gold nanospheres in experiment can be approximately determined as $E_{i}/E_{r}\simeq 0.025$ based on the present theoretical predictions. Interestingly, when $E_{i}/E_{r}$ increases, the variation range of $Q_{total}$ with the glycerol mass fraction gradually narrows, which is physically understandable in that the solid intrinsic damping dominates in the quality factor.

Furthermore, the variations of $Q_{total}$ and $Q_{fluid}$ with the logarithmic radius $\log _{10}[b\,(\text{nm})]$ are depicted in figure 14(b) for the breathing mode of the isotropic gold nanosphere in different glycerol–water mixtures. Note that the loss-to-storage modulus ratio is taken as 0.025 in order to calculate $Q_{total}$ , and the glycerol–water mixtures are described by the compressional non-Newtonian model. It can be found that the qualitative variation trend of $Q_{total}$ is essentially unaltered when incorporating the solid intrinsic damping into the FSI system. Thus, we can refer to the descriptions for the fluid quality factor $Q_{fluid}$ in § 5.3 to explain similar phenomena for $Q_{total}$ and do not repeat that argument here. Again, for a fixed glycerol mass fraction, the variation range of $Q_{total}$ with the nanosphere size is remarkably smaller than that of $Q_{fluid}$ because of the increasing solid intrinsic damping. For different glycerol–water mixtures and various vibration modes, table S2 in SM-V-E provides the theoretically predicted total and fluid quality factors for the reference of experimentalists.