1. Introduction. 2. Concept. reflector. transduce r. node. Kraftmessung an verschiedenen Fluiden in akustischen Feldern

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1 1. Introduction The aim of this Praktikum is to familiarize with the concept and the equipment of acoustic levitation and to measure the forces exerted by an acoustic field on small spherical objects. Acoustic levitation is a technique which enables us to suspend small objects (samples) using acoustic waves. The main advantage compared to other levitation techniques, such as magnetic or electrostatic, is that acoustic levitation is material independent, allowing the handling of a variety of solids and liquids,. Furthermore, it has been demonstrated that can controllably move hovering objects in a 2D plane. 2. Concept The device used in this lab, called single-axis acoustic levitator, consists of two main parts: a) an emitter and a reflector (Figure 1). The reflector is placed opposite of the transducer, leaving a space in-between, where the samples are levitated, called the acoustic chamber. In the axisymmetric levitator, the samples are trapped in one or more distinct points, called nodes. reflector node transduce r Figure 1: Left: Detail of acoustic levitator. Right: Multiple nodes in acoustic levitator Kontakt: Page 1 of 6

2 2.1 Acoustic force The transducer vibrates with a frequency of around khz, emitting acoustic waves in the ultrasonic range (sound frequencies above the audible range). The sound waves travel inside the acoustic chamber until they are reflected back by the reflector. The interactions between the emitted and the reflected acoustic wave create an acoustic pressure distribution inside the chamber with low and high pressure areas. The effect of interacting waves is maximized if a standing wave is established between the reflector and the emitter, as shown in Figure 2. A standing wave is created if the distance H between the emitter and the reflector is an integer multiple of half the wavelength λ. λ H n mit n=1, 2, 3... n 2 (1) The integer n gives us also the number of the pressure nodes (points with minimum pressure) which are created inside the acoustic chamber. The heights H n at which a standing wave is created in the chamber are called chamber resonant heights or we say that the chamber is in resonance. In reality, mainly due to asymmetric intensity of the emitted and reflected waves, the resonant height of the chamber may be found in a non-integer n. The particles are attracted towards the nodes, with the acoustic force having the opposite direction from the gravitational force below the pressure node and the same direction as the gravitational force above the pressure node (Figure 2). For small spherical samples has been shown that the acoustic force F a can be calculated by a potential field U, known as the Gor kov acoustic potential: U2πR p 3ρ c f ρv rms f rms s 2 (2) F a,x U x F a,y U y (3) where ρ rms and v rms are the root mean square values of the acoustic pressure and the particle velocity inside the chamber, R s is the radius of the particle, ρ f and c are the density and the speed of sound in the surrounding medium in our case air and F a,x and F a,y are the acoustic force components in the x and y directions respectively. Such model has been shown to be valid up to R s,<0.1λ, where λ is the wavelength of the acoustic waves. Kontakt: Page 2 of 6

3 We remind that the wavelength can be calculated from frequency f of the sound wave (equal the vibration frequency of the emitter) and the speed of sound c of the medium. Figure 2: Schematic of the pressure distribution P, Gor'kov's potential U and of the acoustic force in the y direction F a,y inside the acoustic levitation chamber. The red stripe shows the low pressure area where the acoustic force counteracts gravity, whereas the green shows the high pressure area where acoustic and gravitational force have the same direction. If the magnitude of the acoustic force is equal to the one of the gravitational force, the sample is levitated. The magnitude of the acoustic force acting on the sample is proportional to the square of the p rms pressure, which in turn is proportional to the amplitude of the emitter vibration velocity V 0. F p 2 V 2 (4) a rms 0 For example, doubling the amplitude of the velocity of the emitter will have a four-fold increase in the acoustic force. The force magnitude is moreover depended on the distance H between the reflector and the emitter. Kontakt: Page 3 of 6

4 2.2 Piezoelectric transducer The source of the acoustic waves is the transducer. The transducer consists of three main parts: a) the piezoelectric elements b) the front mass (emitting the wave) and c) the back mass. The piezoelectric elements are sandwiched between the two masses using a high strength bolt. This design is called Langevin type transducer and ensures a high power acoustic output. If we apply a constant electric potential to the piezoelectric elements, they expand pushing on the front and back masses. If the potential has a sinusoidal alternating form, the whole device is vibrating with the same frequency as the excitation frequency (in that case the frequency of the sinusoidal voltage). emitting surface piezoelectric disks front mass Figure 3: Piezoelectric transducer. back mass Mechanically, the transducer behaves a second order resonator (spring-mass-damper system): s 2ζω s ω 2 s cv (5) 0 0 where s is the displacement on the tip of the transducer, V is the applied voltage, ζ is the damping coefficient, ω 0 is the resonant frequency of the transducer and c is a electromechanical coefficient relating the applied voltage with the force and depends on the properties of the piezoelectric material. Electrically, the model is more complicated resulting in a typical impedance curve as shown in Figure 4. The impedance relates the current flowing through the transducer for a given voltage drop between its electrodes. The current has its peak value for a given voltage amplitude at its resonance frequency and its minimum at its anti-resonance frequency. Kontakt: Page 4 of 6

5 Figure 4: Bode plot of the impedance of a piezoelectric transducer. Note that the resonance frequencies of the mechanical and the electrical systems are the same. One can also infer that the maximum acoustic power occurs in the maximum tip displacement, which takes also place in the resonance frequency. One should not confuse the resonance of the transducer with the resonance of the acoustic chamber (although they are related). Kontakt: Page 5 of 6

6 2.3 Acoustic force measurement Given that we operate our transducer in a constant frequency, preferably near the resonance frequency, the amplitude of the velocity vibration V 0 of the transducer is proportional of the amplitude of current I B flowing through the transducer. The current amplitude can be changed by altering the voltage amplitude V B on the electrodes of the transducer. So, one is able to change the force acting on the samples on the chamber by altering the voltage amplitude. F V 2 I 2 V 2 (6) 0 B B If while a sample is suspended inside the acoustic chamber, we slowly lower the voltage to the transducer, there is threshold where the acoustic field will not be just able to counteract gravity and support the particle. At this point the particle will start falling and the acoustic field force will be almost equal to the force of the gravity. For a particle with given geometry and properties (that is the radius R s, and the density ρ) we are able to calculate the force. 3. Tasks 1. Calculate the distance H between the transducer and the reflector so as to create one node using a acoustic pressure wave of frequency of 25kHz. What should be the distance H to create 2 and 3 nodes? 2. Determine the resonant frequency of the Langevin piezoelectric transducer. a. Connect the transducer to the amplifier. b. Connect the wave generator to the amplifier. c. Connect the voltage and current probes from the transducer to the oscilloscope. d. Connect the laser vibrometer with the oscilloscope and adjust the laser emitter so as to get a good signal. e. Scan the frequencies from 24 khz 26 khz to find the resonance frequency. 3. Measure the acoustic force a. Select a resonant frequency for the transducer (tip: use a frequency a little bit higher than the resonance frequency). b. Calculate the resonant height for 1-node c. Measure the force at the resonant height d. Measure the force for 10 heights ± 10% of the resonant frequency e. Normalize to same velocity vibration amplitude and compare results f. Where do you have maximum force? Why? g. Try to levitate a droplet of water and Ethanol on the maximum force h. How big are the acoustical forces for the two different fluids? Kontakt: Page 6 of 6

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