From the transmission electron microscope images, people knew that the virions of influenza viruses are basically spherical balls packing genomes inside. Since the protein and genome have similar mechanical properties8, for the estimation of dipolar vibration frequencies, we treat the virion as a homogenous sphere.

Dipolar Mode of a Homogeneous Sphere

Due to the spatial confinement, not only electronic but also acoustic energy quantization has been observed in low dimensional systems such as quantum dots and nano-wires. In 1882, Lamb studied the torsional (TOR) and spheroidal (SPH) modes of a homogeneous free sphere by considering the stress-free boundary condition on the surface12. Among these modes, the SPH mode with allows dipolar coupling13 and the corresponding eigenvalue equation can be expressed as14,15:

where , is the spherical Bessel function of the first kind, ω is the angular frequency of the vibrational mode, R is the radius of the nano-sphere, c l and c t are longitudinal and transverse sound velocities respectively. A comparison between the commonly observed breathing mode and the dipolar mode can be found in Supplementary online. the Since the dipolar mode of a nano-sphere cannot be detected by the light scattering experiments16, it was not observed until a previous study of the resonant excitation of dipolar mode through THz wave or microwave excitations17,18 when the core and shell of the nano-sphere have permanent charge separation. Once the resonantly oscillating electric field was applied to the nano-sphere, opposite displacement between core and shell was generated, thus excited the dipolar mode vibrations. Compared with the breathing ( ) and quadrapolar ( ) modes, dipolar mode ( ) is the only SPH mode to directly interact with the EM waves whose wavelength is much longer than the particle’s size. Due to the permanent charge separation nature of viruses, in 2009, dipolar coupling with CAVs is confirmed to be the mechanisms responsible for microwave resonant absorption in viruses by treating spherical viruses as free homogeneous nanoparticles8,9.

Figure 1 shows the simulated displacement field of the dipolar mode (calculated by the finite element method, COMSOL Multiphysics, COMSOL, Inc.) of a homogeneous sphere (mass density and viscoelastic properties are constant throughout the sphere). We define the relative displacement direction of the dipolar mode as the z-direction, which will also be the field direction of the applied EM waves discussed in the next section. By plotting the displacement field of the x-z plane (y = 0) of the sphere, the opposite displacement between the core and shell regions can be clearly observed in Fig. 1(b). Meanwhile Fig. 1 (c) shows the side view of the distortion of the x-y plane of the sphere at different z locations, which concludes that the maximum distortion occurs on the equatorial plane (z = 0) of the sphere. Figure 1(d) shows the top view of the displacement field of the equatorial plane (z = 0). It is interesting to find out that the magnitude of averaged positive displacement (inner region) is 1.27 times the magnitude of the averaged negative displacement (outer region), while positive and negative displacements occupy 42% and 58% area, respectively. Furthermore one can find that the maximum magnitude of the displacement, occurring either at the very center or the outer surface of the equatorial plane, is approximately twice of the averaged magnitude of the displacement.

Figure 1 (a) Schematic showing a homogeneous sphere and applied electric field (b) Displacement field distribution of the x-z plane (y = 0) of the sphere, (c) side view of the distortion of the x-y plane at different z location and (d) top view of the displacement field distribution of the equator plane (z = 0) of the sphere when dipolar resonance mode is excited. Full size image

A Damped Mass-Spring Model

In this work, microwaves were applied to excite the dipolar resonance of the whole virus structure. By exciting the dipolar mode of the nanosphere, core and shell with opposite charge distributions would move in opposite directions and will resonate like a damped mass-spring system17. Our following analysis is similar to the Drude-Lorentz model describing the light-atom interaction, which connects a damped mass-spring system to the quantum-mechanical electronic resonant transitions. In the damped mass-spring system by adopting the reduced mass (m*) of core and shell in the analysis, the relative motion of the displacement can be shown in the following equation:

where z is the relative displacement between the core and shell; b is the damping coefficient, which is related to the surrounding environment; k is the effective spring constant of this system. By assuming z(t) proportional to exp(iωt), one can solve the complex angular frequency of the resonator as:

Therefore the decay rate of the oscillation equals to the imaginary part of the frequency (b/2m*), which corresponds to ω 0 /2Q19:

The intrinsic resonance angular frequency (ω 0 ) of this system is (k/ )0.5. Q is the quality factor of the resonator. From equation (4), stronger damping increased the energy transfer between the resonator and its surrounding environment, which decreases the confinement of the vibration and leads the lower Q. Now we approximate that a spherical virus is like a homogeneous sphere but with opposite and equal charges in the core and shell regions. When the oscillating electric field ( cos ) of microwaves is applied to the system, forced displacements would be induced with the same frequency as the applied microwaves. The equation of motion now needs to include the force induced by the applied electric field (qE), where q is the total amount of charge distributed in the core and shell region of a virus:

We describe the forced displacement as , where A is the amplitude of the forced displacement and is the phase delay between the displacement and the applied electric field. By solving the particular solution of this differential equation, one can obtain the phase delay and the amplitude of the forced oscillating displacement as

and

The instantaneous power absorption of this system is then described as the following equation, where v is the velocity of the oscillating motion17:

By integrating over one full cycle, one can obtain the average power absorption from the system:

Then the absorption cross-section of the virus can be obtained by setting the input power flux as with20

where is the relative permittivity in the system and c is the speed of light in vacuum.

Threshold to Fracture a Virus

With oscillating dipolar vibration to inactivate a virus, the most possible mechanism is to fracture the most outer surface of the equatorial plane (z = 0) due to the location of the maximum distortions, as illustrated in Fig. 1 (c,d). For influenza viruses, this corresponds to the lipid membrane of the envelope. To estimate the maximum induced stress on the equatorial plane, we divide the maximum induced force by the area of the shell region (defined by the moving direction in the approximate model) on the equatorial plane. Following above discussion, we found that the maximum induced stress is twice the average value and the shell region covers 58% of the equatorial plane:

If the required stress threshold to fracture a virus can be obtained, the threshold electric field magnitude of the incident microwaves can also be obtained by using:

Figure 2 shows the threshold magnitude of the incident electric field at different frequencies with different Q based on equation (12) with a fixed threshold value. One can observe that the minimum of the threshold electric field magnitude occurs when the applied frequency is closed to the intrinsic resonant frequency. Moreover cavity quality factor Q plays a major role. By changing the pH value of the solution, charge status of viral surface can be modulated, which affects the Q of the vibration. For example, previous studies indicated that the cavity Q of spherical viruses ranges between 2–10 by raising the pH value of the solution from 5.4 to 7.48. With increased Q, more energy can be confined inside the resonator, which leads to much lower microwave field threshold magnitude at the resonant frequency.

Figure 2 Threshold electric field magnitudes of the incident EM waves to fracture a virus as a function of angular frequency with different Q. Full size image

To experimentally study the efficiency of the SRET from microwaves to CAVs of spherical viruses, influenza A virus subtype H3N2 was used. H3N2 is a subtype of influenza A virus that causes flu. Such viruses can infect birds and mammals and are increasingly abundant in seasonal influenza, which kills an estimated 6309 people in the United States each year, including pneumonia and influenza causes21. Based on previous studies, the averaged mass and the diameter of the H3N2 virus are 161 MDa22 and 100 nm23. Here we approximate the structure of the virus as a nanosphere with a core-shell structure of opposite charge distribution. The shell (90% of the total mass) contains lipid, neuraminidase (NA), hemagglutinin (HA) and M-protein. The core (10% of the total mass) includes RNA and RNP. The reduced mass (m*) of virus is thus 14.5 MDa. From the literature24, force with 400 pN applied on the AFM tip can fracture the lipid envelope. Since the radius of the tip was 30 nm24, the threshold stress to fracture the shell was 0.141 MPa ( ). In order to calculate the threshold magnitude of the electric field to fracture H3N2 virus following equation (12), some important parameters such as q, Q and ω 0 of the studied H3N2 virus has to be obtained by measuring the microwave absorption spectrum of viruses.

As shown in Fig. 3(a) we covered the structure of the coplanar waveguide (CPW) by the microfluidic channel with a 1.25 mm-long sensing zone (L) in order to measure the microwave absorption spectrum of viruses. This microwave microfluidic channel can provide a microwave bandwidth over 40 GHz. The measured results were summarized in Fig. 3(b). As the figure shows, the power absorption ratio (α) by the virus at the resonant frequency (8.2 GHz) was 21% and the Q was only 1.95 by measuring the full width at half maximum of the spectrum. Since the density of viruses (N) in the solution was 7.51014 m−3, experimental absorption cross section of the virus at the resonant frequency can be calculated by the equation below:

Figure 3 (a) Designed CPW circuit for microwave spectrum measured covered with a microfluidic channel. (b) Measured microwave absorption spectrum of H3N2 viruses. (c) Estimated threshold electric field magnitude to fracture the virus as a function of microwave frequency. Full size image

From equation (10), the theoretical absorption cross-section of the virus at the resonant frequency is:

By setting of the PBS at 8.2 GHz as 67.1325, = can be obtained by comparing equation (13) and equation (14).

So far, all parameters for estimating electric field threshold in equation (12) are obtained. By substituting threshold P stress = 0.141 MPa, = , m* = 14.5 MDa, Q = 1.95 and ω 0 = 2π × 8.22 GHz into equation (12), threshold magnitude of electric field to fracture virus at different frequencies of microwave can be calculated. The result is shown in Fig. 3(c). In order to compare with the following inactivation experiments, our estimated threshold magnitude of electric field at 6, 8 and 10 GHz are 103.3, 86.9 and 137.1 V/m, respectively. The minimum threshold occurs close to 8 GHz due to resonance and sufficient internal stress to fracture virus can be expected to be more efficiently generated by weaker electric field.

Based on the IEEE Microwave Safety Standard, the spatial averaged value of the power density in air in open public space shall not exceed the equivalent power density of 100(f/3)1/5 W/m2 at frequencies between 3 and 96 GHz (f is in GHz)26. This corresponds to 115 W/m2 at 6 GHz, 122 W/m2 at 8 GHz and 127 W/m2 at 10 GHz for averaged values of the power densities in air. Assuming all the microwave power in air 100% transmitted into a specimen and by taking the dielectric constant of water 71.92 (6 GHz), 67.4 (8 GHz) and 63.04 (10 GHz)25 for calculation, this safety standard then corresponds to the average electric field magnitude of 101 V/m (6 GHz), 106 V/m (8 GHz), 110 V/m (10 GHz) inside the water-based specimens. It is interesting to notice that the required threshold electric field magnitudes at the resonant frequency (86.9 V/m) to fracture H3N2 viruses as shown in Fig. 3(c) are within the IEEE Microwave Safety Standard (106 V/m), indicating high SRET efficiency, even though the quality factor of the H3N2 virus is low.