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Forces on suspensions or fluids in a standing wave field

When a perfect acoustic standing wave, i.e. one without losses, is set up in a material the energy density per unit volume is constant. That density has kinetic and potential energy components. The kinetic energy has a maximum at the pressure node (where the standing wave acoustic pressure amplitude has a minimum), while the potential energy has a maximum one quarter wavelength away, at a pressure antinode. For almost all of the situations of interest to biologists and biotechnologists the phase of interest is aqueous. The sum of the kinetic and potential energy terms for an inhomogeneity, e.g. cell, particle or bubble, placed in that aqueous phase is different from the energy density of the surrounding fluid. The inhomogeneity (hereafter referred to as a cell) will move spontaneously to a position in the field where the difference between its energy density and that of the surrounding medium is a minimum. Where the standing wave pressure in planes parallel to the transducer is uniform and the gravitational force is ignored then the radiation force DRFa driving a cell of volume V in the axial direction (at right angles to the transducer) towards its potential energy minimum plane is proportional to cell volume and varies with a spatial periodicity of one half wavelength. The force is given by:

DRFa =  -πPo2Vβo

(1)

(2λo)Φ(β,ρ)sin(4πz/λo)

where Po is the sound pressure amplitude, z is distance from a pressure node, and λo is the wavelength of the driving frequency in the liquid. The contrast factor is: Φ(β,ρ) = (5ρc - 2ρo) - βc (2)

(2ρc + ρo) βo

where β is compressibility ρ is density and the subscripts c and o refer to the cell and suspending phases respectively. Biological cells and latex particle for which Φ(β,ρ) is positive, move to the pressure node plane. In systems of practical interest here (frequency of order 1 MHz) the wavelength λo is of the order of 1.5 mm and is much smaller than the typical (order 10 mm) transducer diameter. For this and other reasons pressure gradients exist in the node plane that give rise to a generally weaker lateral component DRFl to the radiation force. This weak radiation force component, together with the microstreaming effects described below, influence the formation of cell aggregates in that plane.

Acoustic streaming

Aqueous solutions or suspending phases also experience direct forces that induce fluid flow. These streaming flows are due to absorption of energy in the bulk phase and energy loss at interfaces, e.g. transducer/fluid or fluid/reflector surfaces. Streaming at solid surfaces can change mass transfer in that region and can induce non-invasive circulation within a closed microchamber.

Interactive forces

Particles in suspension disturb the local pressure environment and as a result the particles experience interactive forces. These forces are pressure and compressibility dependent. The interaction that is important for two equal sized cells of radius r in a pressure node plane is density dependent and attractive and is given, for the case where surface separation is very much smaller than r, by:

Fp = -πr 2(ρc - ρo)2(Po/ρoc)2 (3)

(24ρo)

This force is small compared to, for instance, van der Waals forces at the surface separations (< 30 nm) over which cell surface receptors or crosslinking biomolecules operate.

Cavitation

This general term describes the growth of bubbles within a fluid. Ultrasonic cavitation is well established in biology as a technique for cell disruption. Commercial ultrasonic cell disintegrators operate at frequencies of about 20 kHz. The threshold pressure required to cause a cavitation nucleus to grow increases sharply with frequency around 1 MHz. Applications of standing wave technology to achieve order, structure or controlled flow in a suspension generally avoid cavitation by working at frequencies above 1 MHz.

Thermal effects

Heat is generated as a result of energy loss within the transducer, reflector and the aqueous phase. Practical systems are designed to minimise temperature rise and can, for instance, be significantly less that 0.5 K in ultrasonic traps applied to cell adhesion problems.