Barreras a la participación de los padres

A particle within an acoustic field experiences an acoustic radiation force, which is a non-linear second order effect. Gol’dberg [63] explains that a particle within an acoustically oscillating fluid medium does not necessarily oscillate as one with the medium. Relative movement between the particle and surrounding fluid results in scattering of the acoustic wave and momentum transfer between the fluid and particle. The momentum transfer equates to a force experienced by the particle and is referred to as the acoustic radiation force.

This phenomenon was first recorded by Kundt & Lehmann [1] in the 1870s and recently the scientific community has found new impetus to apply the phenomenon to particle manipulation processes. For example, it is acoustic radiation forces which are used to concentrate particles within the ultrasonic microfluidic separator described later in this chapter.

King [64] derived an expression for the acoustic radiation force acting on incompressible spheres within a plane wave, then Yosioka & Kawasima [65] extended this for compressible spheres. It was shown theoretically that the radiation force acting on a sphere of radiusR due to a plane travelling wave, wave numberk(wherek = 2π/λ) is dependent on(kR)4, which forR λis very small [63] (Gr¨oschl [66] presents a similar argument). However, in a standing wave the force is instead dependent on akRterm and therefore becomes significantly larger. The practical use of radiation forces is therefore limited to standing waves and resonant fields.

For particles suspended within a fluid carrier and in a standing wave propagating in theydirection, the radiation forceFac acts in theydirection and is also a function of theyposition. It is quoted by Gr¨oschl [66] and others in a form similar to the following:

hFac(y)i = Fac0sin(2ky), (2.8)

where Fac0 =

4

3πhε¯ikR 3_φ.

In the above equationhε¯i is the time averaged energy density which can also be expressed as a function of the acoustic pressure amplitude. kis the wave number andφis the acoustic contrast factor, defined as:

φ= 5ρp−2ρf 2ρp+ρf −cf 2_ρ f cp2_ρp. (2.9)

whereρpandρf are the particle and fluid densities, andcpandcf represent speed of sound for the particle and fluid mediums.

Gor’kov [67] derived an expression for the radiation force acting on a particle in an arbitrary field. The following is taken from Gr¨oschl [66]:

hFac(x, y, z)i=−∇hφG (x, y, z)i, (2.10) where hφG (x, y, z)i=−V " 3(ρp−ρf) 2ρp+ρf hE¯kini − 1− cf2ρf cp2ρp ! hE¯poti # (2.11)

whereV is particle volume,hE¯kiniandhE¯potiare the time averaged kinetic and potential energies at a point in the field and φG _{is the gradient of radiation force potential. These equations are} generally true in the case ofkR1[63].

Referring to equation (2.11), if the velocity and pressure fields are known it is possible to derive the time averaged kinetic and potential acoustic energy densities. For a progressive wave the time averaged forcehFacitends to zero, as the gradients of both the time averaged pressure and velocity fields are small, therefore leading to negligible displacement of particles. In contrast, in a plane standing wave the expression simplifies to equation (2.8) (part of this derivation is given by Yasuda [68]) and is a 1-dimensional description of the radiation force with a rigid boundary existing at y= 0.

To understand the physical influence of the radiation force on particles, figure 2.9 illustrates the spatial relationship between the acoustic pressure field of a standing wave and radiation force, shown over a distance ofλ/2. It can be seen from the diagram and equation (2.8) that the radiation force varies at twice the spatial frequency of the acoustic wave and can also act in a positive or negativeydirection.

Figure 2.9 illustrates the case forφ >0, typical for solid particles, where the radiation force causes particles to converge towards the nearest pressure node which are sites of low potential energy, the force also being a function of position in the wave. For example, particles positioned in region A experience a downward force, away from the pressure anti-node and towards the pressure node, at which point the force reduces to 0 and, assuming no other disturbances to the particles, they come to rest. Particles in region B however experience an upward force, again away from the anti-node and towards the pressure node. In this system, particles are repelled from the anti-nodes and attracted to the nodes where the particles agglomerate.

Figure 2.9: Acoustic force in a standing wave.

Conversely, for particles whereφ < 0 such as for benzene liquid phase droplets in water [69], equation (2.8) shows that these particles are instead drawn to the pressure anti-node. A more complex case arises for gas bubbles which although typicallyφ < 0 in their equilibrium state, move to either the pressure node or anti-node depending on bubble size [70, 71]. This is associated with the bubble itself resonating and the resulting periodic change of the gas density. For the application with which this thesis is concerned, solid particles are used resulting typically inφ >0 and movement towards the pressure nodes.

Measurement of Acoustic Radiation Force

Generally, literature investigating acoustic radiation force experimentally have demonstrated a good level of agreement with the theory presented here [72, 73]. However, some discrepancies with the theory have been discussed, for example, by Yasuda & Kamakura [74] who observed that for particles of diameter less than 5µm, the linear dependence of the acoustic force on the volume, V, of a particle begins to fail. This discrepancy was explained by considering the existence of a shell around the particle increasing the apparent radius of a particle, R0, one possible cause being a viscous fluid boundary layer. The thickness of the boundary layer,δ, based on Yasuda and Kamakura’s paper, can be calculated as follows:

δ=

s

2µ ρfω

(2.12)

Based on this proposed explanation, for an operating frequency of approximately 3MHz (ω =

2πf) and fluid parameters based on those of water, the boundary layer thickness of a particle is ∼0.3µm, where the apparent radiusR0 = R+δ. This example translates to a 50% increase in radiation force for a2µm particle, down to a 9% increase for a10µmradius particle. However, this hypothesis does not consider streaming flows which are commented on in section 2.3.2, and which provide another explanation for the behaviour of small particles.

Secondary Radiation Force

Particles within an acoustic field experience forces other than the primary acoustic radiation force. The primary radiation force as previously discussed is a result of the interaction between the os- cillating fluid and scattering from particles, however, a particle can also be influenced by the scat- tered field from a neighbouring particle [66, 75]. The radiation force resulting from these particle interactions is known as Bjerknes force or secondary radiation force (although in the literature this is occasional used as a collective term to describe lateral acoustic forces resulting from both particle-fluid and particle-particle interactions). The magnitude of the particle interaction force is a function of the acoustic velocity and pressure profiles, such that particles aligned with the direc- tion of sound propagation will experience a repulsive force whilst those aligned normal to sound propagation experience an attractive force.

The force is also a function of the reciprocal of particle spacing and therefore becomes negligible for a low concentration suspension where particle spacing is large. However, as particles begin to agglomerate onto a plane due to the primary radiation force reducing particle spacing, the secondary radiation forces increase. This means that for particles gathered at the pressure node, they are also forced together laterally by Bjerknes forces into clusters or as a membrane on the nodal plane. As these secondary radiation forces are still small relative to the primary radiation force, it takes a significant time to form particle clusters. Therefore secondary radiation forces have been disregarded in some analyses of flow-through devices where particles are exposed to the acoustic field for a short, finite length of time just sufficient to move particles to the nodal planes, or where particle concentration is low [76].

Figure 2.10: Lateral radiation forces acting on particles.

Lateral Radiation Forces

The preceding pages have illustrated the effect of acoustic radiation force in terms of a simple acoustic field giving rise to an axial radiation force, aligned in the direction of the acoustic prop- agation. However, in practical systems lateral variations in the acoustic field typically occur and give rise to lateral radiation forces [66, 77].

To demonstrate the movement of particles under these forces, figure 2.10(a) illustrates the motion of particles within a standing wave, initially subjected to the axial primary radiation force. For a positive value of Fac0 the particles are forced towards the pressure node until they lie in the nodal plane depicted in (b). Similarly, variations in the acoustic field within the nodal plane create a radiation force potential, φG_{, which creates a radiation force as described by equation (2.11)} [78]. This force is perpendicular to the axis of the sound wave and, depending on the nature of the acoustic field, moves particles towards a single point (c) or into striations as observed during preliminary experiments with the ultrasonic separator. The source of these lateral variations are due to, for example, the cavity boundaries and divergence of the wave [66].

Schram [78] comments that depending on the geometry of the resonating system and acoustic beam, lateral radiation force could be comparable to the axial component. Various experimental investigations have been undertaken to measure or characterise lateral radiation force [77, 79, 80, 81]. For example, Whitworth & Coakley observe the lateral movement of particles within a cylindrical cavity with results comparing well with the analytical description of the acoustic field. Woodside et al. [80] measure lateral forces within a rectangular resonator and compare this

with the axial radiation force, revealing 100-fold difference between the axial and smaller lateral forces. With a similar system, Woodside et al. [81] also measure the amplitude of displacement of the reflector and transducer surfaces and link lateral variations in displacement amplitude to the lateral movement of particles within the device, with the resonant patterns on the reflector surface explained in terms of acoustic shear waves. More recently, Lilliehorn et al. [82] use the near-field interference pattern of an acoustic field to give rise to lateral variations and high lateral forces to trap particles in 3 dimensions.

In summary, lateral radiation forces are shown in the literature to be caused by non-uniformities in the acoustic field which themselves may be generated by a variety of resonant effects, including the fluid cavity boundary and geometry, near-field interference and boundary shear-waves. Depending on the design of the resonator and cause of the lateral variations, lateral radiation forces may become comparable to the axial component of radiation force.