# Engineering Acoustics/Acoustic Levitation

## Definition

Acoustic levitation employs sound radiation to lift objects. It mostly
deals with non-linear phenomenon (since the resulting force on the
object is due to non linear properties of wave motion).

## Motivation behind developing an acoustic reactor

The force generated due to acoustic radiation pressure is generally much
larger than force of electromagnetic radiation pressure which makes the
study of these forces interesting and noteworthy.

Secondly, this phenomenon will allow successful containerless
experiments. The importance of such studies is illustrated by the
following:

Kinetic studies can be classified into two categories:

1.  The first includes material fixed to the walls.
2.  The second includes the flow of particles into and from an apparatus

The drawback of existing methods is that only one type of particle can
be used. Consequently, the behavior reported isn\'t accurate (since the
walls in the first case and the surrounding particles in the second case
can have an effect on the behavior under study).

This elimination of walls can provide further insight by discarding
supports in addition to reducing the interactions with other particles
(e.g.: by handling a single bubble).

One way to achieve this airborne application is by employing a
fascinating application of acoustics, namely acoustic levitation which
involves levitating objects using sound radiation.

Applications of this phenomenon and the corresponding technology can
include material processing in space without using any containers. This
may be particularly useful in the study of materials that are extremely
corrosive.

Moreover, sonoluminescence and acoustic
cavitation encounter this acoustic force.

Other applications can include measuring densities and analyzing fluid
dynamics in which surface tension plays an important role. Lastly,
acoustic positioning is another potential application.

-   Discovery
    News
    lists an interesting application of acoustic levitation.
-   Acoustic Levitation on
    Mars illustrates an
    adventurous application of this technology.

## Components of an Acoustic Reactor

!Figure 1: Schematic of the set
up{width="300"}
A simple acoustic reactor requires a:

-   A transducer to generate the desired sound waves. These transducers
    usually generate intense sounds, with sound pressure levels greater
    than 150 dB.
-   A reflector

In order to focus the sound, transducers and reflectors in general have
concave surfaces. The reflection of longitudinal sound waves off the
reflector leads to interference between the compressions and
rarefactions. Perfect interference will result in a standing acoustic
wave, i.e., a wave that will appear to have the same position at any
time.

With this simple arrangement of transducer and reflector, one can
achieve stable levitation but cannot steer the sample. To do so, Weber,
Rey, Neuefeind and Benmore have described an arrangement in their paper
that describes the use of two transducers. These transducers adjust the
location by altering the acoustic phases (which is carried out
electronically).

## Single Bubble Sonoluminescence

This phenomena occurs when a single bubble encounters a non linear
dynamic, namely, rapid compression of bubble preceded by an expansion
which takes place slowly. When the bubble is compressed rapidly, it can
get so hot that it emits a flash of light

!Figure 2: Sonoluminescence -
mechanism{width="500"}

## Theory

(source: Theory of long wavelength acoustic radiation pressure by
Löfstedt and Putterman)

Starting with the integral form of conservation of momentum,

$$\frac{\partial \rho v_i}{\partial t}+ \frac{\partial \Pi_{ij}}{\partial r_j}=0$$

$$\int_{v}\frac{\partial \rho v_i}{\partial t}+ \int_{S_o}\Pi_{ij} dS_j+\int_{S_K}\Pi_{ij} dS_j=0$$

where

`<big>`{=html}$\Pi_{ij}$`</big>`{=html} is the stress tensor,

`<big>`{=html}$\rho, v$`</big>`{=html} are the local fluid density and
velocity,

`<big>`{=html}$S_o$`</big>`{=html} the surface of the object (at time
t),

`<big>`{=html}$S_K$`</big>`{=html} a surface far from the object, and

V the volume bounded by these surfaces.

Using the relation,

$$\int_v \frac{\partial \rho v_i}{\partial t} dr + \frac{d}{dt}\int\rho v_i dr - \int_{S_o} \rho v_i v_j dS_j = 0$$

$$\frac{d}{dt} \int \rho v_i dr + \int_{S_o} (\Pi_{ij} - \rho v_i v_j) dS_j + \int_{S_R} \Pi_{ij} dS_j = 0$$

Time average of this equation gives an expression for the force on a
moving sphere

$$\langle F_i \rangle  =  \langle \int_{S_o} (\Pi_{ij} - \rho v_i v_j) dS_j \rangle  =  - \int_{S_R} \langle \Pi_{ij} \rangle dS_j = 0$$

Assuming an ideal fluid,

The Galilean invariant contribution to the stress tensor is

:   `<big>`{=html}`<big>`{=html}`<math>`{=html}

\\Pi\_{ij}-\\rho v_i v_j = p\\delta\_{ij}
`</math>`{=html}`</big>`{=html}`</big>`{=html}

and

$$p = - \rho_{eq} \frac{\partial \phi}{\partial t} - \rho_{eq} v^2/2 +  \frac{\rho_{eq}}{2c^2} \left({\frac{\partial \phi}{\partial t}}\right)^2$$

here, `<big>`{=html}$v=\nabla \phi$`</big>`{=html}

`<big>`{=html}$\rho_{eq}$`</big>`{=html} represents the equilibrium
density

`<big>`{=html}$c$`</big>`{=html} denotes the speed of sound

Acoustic radiation force on an object in an ideal fluid is

```{=html}
<center>
```
$\langle F_i \rangle = - \int_{S_R} {[- \frac{\rho \langle v^2 \rangle}{2} +  \frac{\rho}{2c^2} \langle{\frac{\partial \phi}{\partial t}}^2\rangle + \rho \langle v_i v_j \rangle] dS_j}$

```{=html}
</center>
```
### Multipole Expansion of acoustic radiation force

Consider the linear wave equation,

$$\frac{\partial^2 \phi}{\partial t^2} - c^2 \Delta^2 \phi = 0$$

where

:   `<big>`{=html}`<big>`{=html}$\phi = \phi_i + \phi_s$`</big>`{=html}\
    `</big>`{=html}

here `<big>`{=html}$\phi_i$`</big>`{=html} is given by transducer,

`<big>`{=html}$\phi_s$`</big>`{=html} is given by the corresponding
boundary condition at the object where s stands for \'scattered\'

$\phi_s = Re \sum_{n=0}^\infty B_n h_n (k r)P_n (cos\theta) e^{-i\omega t}$

`<big>`{=html}$h_n$`</big>`{=html} here are the outgoing spherical
Hankel functions

`<big>`{=html}$P_n$`</big>`{=html} here are the Legendre polynomials

`<big>`{=html}`<big>`{=html}$\omega = k c$`</big>`{=html}`</big>`{=html}

k: wave number of the sound field imposed on the object

As r approaches infinity,

$\lim_{r \to \infty} \phi_s \to Re \frac{e^{i (kr - \omega t)}}{kr} \sum_{n} (-i)^{n+1} B_n P_n (cos\theta)$

For standing waves,

:   `<big>`{=html}`<big>`{=html}`<math>`{=html}

\\phi_i = Re A sin(kz) e\^{-i\\omega t}
`</math>`{=html}`</big>`{=html}`</big>`{=html}

Thus by computing $\phi$ and using the result in the expression for the
radiation force we get

```{=html}
<center>
```
`<big>`{=html}`<big>`{=html}$F_z = 2\pi\rho A[coskz_o(Im B_o)-sinkz_o(Im B_1)$`</big>`{=html}`</big>`{=html}
\...\...\...\...\...\...\...\...\...\...\...\...\...\..... (1)

```{=html}
</center>
```
### Radiation Force on a spherical object

Consider a spherical body with radius $R_o$ and density ρ~o~

The wave equation inside the sphere is given by:
$:\frac{\partial^2 \Phi}{\partial t^2}-
c_o^2{\nabla^2 \Phi_o} - \zeta_o \rho_o \frac{\partial \nabla^2 \Phi_o}{\partial t} = 0$

where

`<big>`{=html}$\Phi_o$`</big>`{=html} is the velocity potential

`<big>`{=html}$c_o$`</big>`{=html} is the speed of sound in the sphere

`<big>`{=html}$\zeta_o$`</big>`{=html} characterizes the damping in the
sphere

(Effects due to thermal conductivity and shear viscosity are neglected)

The solution to this equation is given by:

$:\Phi_o = Re \sum_{n=0}^\infty C_n j_n (k_o r)P_n (cos\theta) e^{-i\omega t}$

where `<big>`{=html}$j_n$`</big>`{=html} are spherical Bessel functions

`<big>`{=html}$k_o$`</big>`{=html} is complex (because of dissipation)

`<big>`{=html}$k_o = k_o +i \alpha_o$`</big>`{=html}

where

`<big>`{=html}$k_o = \omega/c_o$`</big>`{=html}

`<big>`{=html}$\alpha_o = k_o^2 \zeta_o/2c_o \rho_o$`</big>`{=html}

Here,`<big>`{=html}$\alpha_o$`</big>`{=html} is the attenuation
coefficient of sound in the sphere

The boundary conditions are given by

`<big>`{=html}`<big>`{=html}$\rho \Phi (R_o) = \rho_o \Phi_o (R_o)$`</big>`{=html}`</big>`{=html}

At r = `<big>`{=html}$R_o$`</big>`{=html}

$\frac{\partial \Phi}{\partial t} = \frac{\partial \Phi_o}{\partial t}$

To satisfy these conditions, the incident wave is expanded using
spherical harmonics

$sin(kz) = sin(k z_o) \sum_{n=0}^\infty a_{2n} (kr) P_{2l}  cos\theta + cos(k z_o) \sum_{n=0}^\infty a_{2n+1} (kr) P_{2l+1}     cos\theta$

where

$a_{2l} = \frac{4l+1}{2}. \int_{-1}^1 cos krx P_{2l} (x) dx

a_{2l + 1} = \frac{4l+3}{2}. \int_{-1}^1 sin krx P_{2l+1} (x) dx$

Using the above relations, one can compute $a_o$

$a_o = \frac{sin kr}{kr}.$

When `<big>`{=html}$kR_o << 1$`</big>`{=html}

$sin kz = sin kz_o [ 1 - \frac{1}{2}. k^2 r^2 cos^2 \theta + ...] + cos kz_o [kr cos \theta - ...]$

The boundary condition for the case of a standing wave can be deduced as
follows:

For monopole term,

$\rho A sin k z_o - \rho B_o \frac{i e^{ikR_o}}{kR_o} = \rho_o C_o j_o (x_o)$,

and

$- A (\frac{k^2 R_o}{3} sin k z_o + B_o e^{ikR_o}(\frac{1}{R_o} + \frac{i}{(kR_o)^2}) =  C_o k_o j^{'}_o (x_o)$

For dipole term

$\rho A (k R_o) cos k R_o - \rho B_1 e^{ikR_o} (\frac{1}{kR_o} + \frac{i}{(kR_o)^2}) = \rho_o C_1 j_1 (x_o)$,

and

$A k cos k z_o + B_1 e^{ikR_o} (\frac{-i}{R_o} + \frac{2}{kR_o^2}+ \frac{2i}{k^2 R_o^3}) = C_1 k_o j^{'}_1 (x_o)$,

where

`<big>`{=html}$x_o = k_o R_o$`</big>`{=html}

`<big>`{=html}$B_1$`</big>`{=html} and
`<big>`{=html}$B_o$`</big>`{=html} can be obtained as functions of A

Using these relations for `<big>`{=html}$B_o$`</big>`{=html} and
`<big>`{=html}$B_1$`</big>`{=html} in the radiation force expression we
get:

Thus the radiation force on a sphere is given by

`<big>`{=html}`<big>`{=html}$F_{z} = - \pi k^3 R^3 A^2 sin 2kz_o Re [f_o + f_1]$`</big>`{=html}`</big>`{=html}

where
$f_o = \frac{(1/3)(\rho_o/\rho) k^2 R_o^2 b_o (x_o) + 1}{k^2 R_o^2 [1 + (\rho_o/\rho) b_o (x_o)+ ikR_o]}$

$f_1 = \frac{(\rho_o/\rho) b_1 (x_o) - 1}{2(\rho_o/\rho) b_1 (x_o)+ 1}$

with

$b_o (x_o) = \frac{j_o (x_o)}{x_o j^'_o (x_o)}$

$b_1 (x_o) = \frac{j_1 (x_o)}{x_o j^'_1 (x_o)}$

If we neglect damping, assume $x_o << 1$ and assume the sphere is
incompressible (i.e. $c_o$ approaches infinity), then the radiation
force simplifies to:

```{=html}
<center>
```
$F_{z} = - \pi k^3 R_o^3 A^2 sin 2kz_o \rho_o {5 \rho_o-2 \rho\over 3(2\rho_o + \rho)}$

```{=html}
</center>
```
This expression for radiation force (in a standing wave field) was
initially derived by King. Note that the radiation force is directly
proportional to the cube of the radius and directly proportional to the
velocity amplitude.

### Assumptions

1.  kR \<\< 1, i.e., the wavelength of the sound field is much larger
    than the dimension of the sphere.
2.  incompressible object (used by King, although Gorkov has derived
    results that permits finite compressibility of the sphere)

Sphere is suspended when sum of the forces acting on it equals zero,
i.e., when force due to gravity balances the upward levitation force.

As a result, the object is attracted to regions of minimum potential
energy (pressure nodes). Antinodes are regions experiencing high
pressures.

To ensure the generation of a standing wave, the transducer must be
placed at a certain distance from the reflector and a particular
frequency should be used to get satisfactory results. This distance
should be a multiple of half the wavelength of the sound produced to
make sure the nodes and antinodes are stable.

Secondly, the direction of the force exerted by the radiated pressure
due to the sound waves must be parallel to the direction of gravity.

Since the stable areas should be large enough and able to support the
object to be levitated, the object\'s dimensions should lie between one
third and one half of the wavelength. It is important to note that the
higher the frequency, the smaller the dimensions of the object one is
trying to levitate (since wavelength and frequency are inversely
proportional to each other)

The materials of the object is important too, since the density along
with the dimensions will give the value for its mass and determine the
gravitational force and consequently whether the upward force produced
by the pressure radiation is suitable.

Another characteristic important when talking about material properties
is the **Bond number** which is important when dealing with drops of
fluid. It characterizes the surface tension and size of the liquid
relative to the fluid surrounding it. The lower the Bond number, the
greater the chances that the drop will burst.

Finally, to achieve such high pressures (that can cancel the
gravitational force), linear waves are insufficient. Therefore,
non-linear waves play an important role in acoustic levitation. This is
easily one of the reasons why the study of acoustic levitation is
challenging. Nonlinear acoustics is a field that deals with physical
phenomena difficult to comprehend. Based on experimental observations,
heavy spheres incline to velocity antinodes, light particles are closer
to the nodes.

## Other effects on levitation force

Temperature, pressure, fluid medium characteristics (density, particle
velocity) affect the levitation force. It is important to remember that
the medium changes as conditions change. The fluid medium consists of
reactants and products that change with reaction rate.

Thus, consequently the levitation force is affected. To compensate for
medium changes, resonance tracking system can be employed (which helps
to maintain stable levitation of the particle under study)

## Design considerations

The sphere or particle under study should experience a lateral force
which will act as a positioning force (along with the more obvious
vertical levitating force) Rotation of the sphere about its axis will
ensure uniform heating and stability.

## Using non-spherical particles

When levitating non-spherical particles, the largest cross section of
the object will end up aligning itself perpendicular to the axis of the
standing wave.

## Traveling vs Standing waves

King discovered that the radiation pressure exerted by a standing wave
is much larger than the pressure exerted by a traveling wave (which has
the same amplitude as the standing wave)

This is because the pressure exerted by a standing wave is due to the
interference between the incident and scattered waves. Pressure exerted
by a traveling wave is due to contributions from scattered field only.

## References

1.  Theory of long wavelength acoustic radiation pressure by Löfstedt
    and Putterman
2.  Development of an acoustic levitation reactor by Cao Zhuyoua, Liu
    Shuqina, Li Zhimina, Gong Minglia, Ma Yulongb and Wang Chenghaob
3.  HowStuffWorks

## Useful Sites

1.  Spherical Hankel
    Functions
2.  Legendre
    Polynomials
3.  Multipole expansion
