School of Food Science and Nutrition
FACULTY OF MATHEMATICS AND PHYSICAL SCIENCES
Malcolm J. W. Povey
December 9 th to December 11 th 2014

Reproduction
You may freely use this presentation.
You may reproduce the material within on condition that you reference the original source(s).
The author asserts his moral and paternity rights regarding the work.

Welcome
Welcome to the School of Food Science and
Nutrition
This course addresses the fundamental physical questions needed to understand a range of practical applications of ultrasound. Many of these applications have been developed here.
There are no course pre-requisites, apart from an interest in ultrasound as a practical tool for the study of materials. Some of you may feel that I am teaching my grandmother to suck eggs.
Please be patient, sucking eggs is not as easy as it looks. Not everyone knows how to do it.

What’s special about soft solids?
Very many foods are ‘soft’ solids
Soft solids have the properties of BOTH fluids and solids
Soft solids show time dependent elastic properties on human time scales.
The elastic properties of soft solids are time dependent.

The Beginnings
1826, the first determination of the speed of sound in water http://en.wikipedia.org/wiki/Jacques_Charles_Fran%C3%A7ois_Sturm

You need the proper tools to understand Sound
Ultrasound transduction system
Digital oscilloscope
Microphone

Recommended books
Keywords: Ultrasound, ultrasonics, ultras*, acoustic*, sound, propagation, scattering, diffraction, interference,
Ultrasonic techniques for fluids characterization, Malcolm Povey, Academic Press, San
Diego, 1997

Metaphors
Use light as a metaphor
Here the suns rays are scattered from the back of the cloud, creating miniimages of the sun. The cloud absorbs the light, with darkness at the front and light at the back.
These are called anticrepuscular rays.

Sound pulse in air
Restored Expanded Compressed
Undisturbed
X
Amplitude and direction of particle displacement
Pressure variation

A shear pulse
Restored http://www.acoustics.salford.ac.uk/feschools/waves/wavetypes.htm
Undisturbed
X

Surface waves
Lamb wave in a plate
Diving grebe (wikipedia)

Piston source
Region of confusion

The density of phonon modes
5
4
3
2
1
0
0 0.2
0.4
0.6
Frequency / 10
13
Hz
0.8
1
A phonon is a quantum of sound.
Heat is composed of phonons, so all heat is made up of sound waves. But most of them are very high frequency.

Light and ultrasound
Ultrasound
Transducers are phase sensitive
Wavelength between m and
 m
Frequency between 0.1 and 10 13 Hz
Coherence between pulses
Visible Light
Transducers are phase insensitive
Wavelength between 0.5 and 1
 m
Frequency between 3

10 16 and 6

10 16 Hz
No coherence between pulses
Responds to elastic, thermophysical, and density properties
Particle motion parallel to the direction of propagation; no polarization
Propagates through optically opaque materials
Responds to dielectric and permeability properties
Field displacement perpendicular to direction of propagation; polarization is therefore possible
Sample dilution is normally required

The adiabatic approximation
Restored Expanded Compressed
Undisturbed
Heat flow restricted to a small region of a half wave
Amplitude and direction of particle displacement
X
Pressure variation

Mathematical description
Period T=1/f
Time v
 f

Wavelength

Distance

Decaying wave

 x

Sound velocity measurement
Source
Transducer
32.5 mm t=0
 s t=50
 s
Velocity in sample equals distance / time,
32.5 mm / 50
 s = 1500 m s-1 for single transit and
75 mm / 100
 s = 1500 m s
-1
for a single reflection.
Receiver transducer or reflector

Group and Phase velocity v
Group velocity
 d
 dk k is called the wave number,
λ is the wavelength k 
2 

Source
Transducer
32.5 mm
This is the velocity of the wave envelope t=0
 s v p e.g ocean waves s
Velocity in sample equals distance / time,
32.5 mm / 50
 s = 1500 m s-1 for single transit and
75 mm / 100
 s = 1500 m s
-1
for a single reflection.
Receiver transducer or

 k
Phase velocity is the speed of a given frequency component within the wave

Attenuation

 exp
   x
 x

Velocity and attenuation k k k

 
  k
  i k
 v
 p
This is called the wave VECTOR because it comprises two numbers, the first one is sometimes called the
‘real’ number and the second the
‘imaginary’, because it is multiplied by the square root of minus one.

1
Attenuation coefficient

Velocity, phase and attenuation
Particle displacement
  
0 exp
 
0 exp
 i k
 x
  x

 
0 exp
 i v p
  x p
Instantaneous sound pressure
 p
0 exp
 i (
 t
 k x

Maximum sound pressure p
 p
0 exp
 i (
 t
 k ' x )
 
  x


Definitions of attenuation p

2
1 x ln
~


0
  
1
10 log x



 

0


Neper, x = 1 meter.
dB, x = 1 meter

UVM
Pulser
Timer start
Timer stop
Thermostated water bath
Magnetic stirrer
Temperature
Probe
Four-wire Resistive
Temperature Device (RTD) mV output to computer
Time-offlight reading to computer
Computer
Accepts time of flight and RTD mV and produces velocity of sound and temperature readings.

Waveforms and group velocity
Trigger Pulse
Transducer(1)
140
 s repetition period
Trigger Pulse
Transduce r (2) first echo second echo
30
 s ultrasound propagation delay
80
 s second echo propagation delay

Transducer construction
Silver loaded epoxy electrode
Electrical connection welded to transducer electrode
Tungsten loaded epoxy backing plate
Piezo-ceramic transducer disk
Wear plate
Silver loaded epoxy ground plane

Impedance Z
Z 
 p
 
 
 k
  v
In words: The impedance is the ratio of the pressure change resulting during the passage of the wave to the particle velocity. This approximates to the product of the density times the speed of sound.

Reflection and transmission
Incident (

I
)
Transmitted (
 t
)
Transmission coefficient
 

 i t 
Z
1
2 Z

1
Z
2
Reflected (
 r
)
Material Impedance Z
2
Reflection coefficient
 

 i r 
Z
1
 Z
Z
1
 Z
2
2
Material Impedance Z
1

Reverberation
Incident acoustic field
2t
Reverberations
+2t
+4t
+6t
+
+
+
+
Glass slide suspended in water

Coupling and buffering
Sample
Buffer rod Buffer rod
Piezo-ceramic disk transducers

Table 2-1 Typical power levels and other propagation parameters for ultrasound propagation in water at 1 MHz and 30

C .
Power levels and propagation parameters at 1 MHz and 30 o C .
f T0 p0 I  p
0 s  
(MHz) ( o K) (MPa) (kW m -2 ) (MPa) x 10 -6 (nm) (mm s -1 )
 ’
(km s -2 )
1 303 0.1 0.1 0.017 7.6 1.8 11.5 72
1 303 0.1 10 0.17 76 18 115 720
1 303 0.1 1000 1.7 760 180 1150 7200
Z
(MPa s m -1 )
1.47
1.47
1.47
 T  ’/v
(mK) x 10 -6
.38 7.6
3.8
38 760
76
Here f is frequency (in MHz), T0 , absolute temperature (K); P0 , absolute pressure (MPa); I , intensity
(kW m-2);
 p, pressure change from P
0
owing to the passage of ultrasound (MPa); s , the condensation (
=[

0-

]/

0);

0
, the static density (kg m-3);

, the instantaneous density (kg m
-3
);

, the particle displacement (nm);
 ‘
, the particle velocity (mm s-
1
);
 ‘’
, the particle acceleration (km s-2);

T , the temperature change owing to the passage of the ultrasound (K); Z ( =

P /
 ‘ =  vl ), the specific acoustical impedance (Pa s m-1); and v the velocity of a compression ultrasound wave. From Povey and McClements
(1988

Axial intensity
1
Near field
Focus
Far field a 2 /3
 a 2 /2
 a 2 /

0
0 1
X/a
Point spread function courtesy of Nick Parker
2

Wavefronts and phase
Pressure
Advancing wavefront
Lines of constant phase
Distance

Fraunhofer diffraction
Pressure
Advancing wavefront
Lines of constant phase
Distance

Incoherence
Pressure
Distance
The wave front can break up like this due to diffraction and scattering.
The transducer will not detect the wave front because the phase variation across the transducer face sums to zero.

Trigger errors

The wood equation
Bulk modulus v 
B


1

Adiabatic compressibility
Density

Sound velocity in air/water mixtures

Urick equation
Phase volume of jth phase v 
1

,  
 j
, 
 j
  j
2
   
1
,
 
2
   
1

Velocity of sound in water
The Marczak polynomial is recommended for calibration purposes
Marczak c = 1.402385 x 103 + 5.038813 T - 5.799136 x
10-2 T2 +3.287156 x 10-4 T3
- 1.398845 x 10-6 T4+2.787860 x 10-9 T5
Marczak (1997) combined three sets of experimental measurements, Del Grosso and
Mader (1972), Kroebel and Mahrt (1976) and
Fujii and Masui (1993) and produced a fifth order polynomial based on the 1990
International Temperature Scale. Range of validity: 0-95OC at atmospheric pressure
W. Marczak (1997), Water as a standard in the measurements of speed of sound in liquids J. Acoust. Soc. Am. 102(5) pp 2776-
2779.
N. Bilaniuk and G. S. K. Wong (1993), Speed of sound in pure water as a function of temperature, J. Acoust. Soc. Am. 93(3) pp
1609-1612, as amended by N. Bilaniuk and G.
S. K. Wong (1996), Erratum: Speed of sound in pure water as a function of temperature [J.
Acoust. Soc. Am. 93, 1609-1612 (1993)], J.
Acoust. Soc. Am. 99(5), p 3257. C-T Chen and
F.J. Millero (1977), The use and misuse of pure water PVT properties for lake waters, Nature
Vol 266, 21 April 1977, pp 707-708.
V.A. Del Grosso and C.W. Mader (1972),
Speed of sound in pure water, J. Acoust. Soc.
Am. 52, pp 1442-1446.

Compressibility of water v 
 .
.
 .
T  .
T 2
 6 T 4  .
 9 T

5
.
 .

 4 T 3
 6 ( p
0
 10 5 )

Sound velocity in margarine

Dependence of sound velocity on solids d) v for 10% w/w oil c) v for 60% w/w oil b) v for 80% w/w oil a) % solids
1 v
2
 i n 

1
 i v i
2

Modified Urick Equation v
1
2
 v
1
2
1

1    2

 


 a 2

 a 1
 a 1

2
 

1
1


R 





2

2
C p 2

1

1
C
  (   1 )
 p

1
2

1

1
C

1
C
C p 1 p 2 p 1




R 2
 

 a 2

 a 1
 a 1  
 

2
 

1
1

2


2
 
1

2
3 
1
2

Partial molar volume

Acoustic scattering
Basic science
Molecules as particles
LFPST
Soft solids
Viscosity measurement
Bat sounds

The ‘classical’ model for attenuation
Bulk viscosity - ratio of specific heats - thermal conductivity
 cl
 
2


2
 v v
0
3


4
3



B v

  
1
 
C
P
 v


Attenuation - radial frequency - density – velocity - shear viscosity

Underlying physics
Conservation of momentum -Newton’s second law, force is mass (m) d
F

Conservation of mass dt
Together conservation of momentum and conservation of mass give rise to the Navier-Stokes equation for fluids. In soft solids an even more complicated relationship exists due to time dependent shear and compressibility.
Conservation of energy
Second law of thermodynamics

Attenuation in water
Total attenuation
10000
1000
100
10 y = 0.200x
2 + 1.361x - 17.93
R² = 1
Experiment
Classical continuum theory y = 0.218x
2 + 1E-12x - 4E-11
R² = 1
100 f /MHz

Data for water
[°C] [Pa·s]
10 0.00130
20 0.00100
25 0.00089
30 0.00080
40 0.00065
50 0.00054
60 0.00046
70 0.00040
80 0.00035
90 0.00031
100 0.00028
Shear viscosity
Attenuation data
Density of water
Frequency
Speed of sound
Ratio of specific heats
Thermal conductivity

Bubbles
On Musical Air
Bubbles and the Sounds of Running
Water,
Minnaert, M.,
Phil. Mag., 1933.

Surface active and microbubbles
Key authors
Andrea Prosperetti
Gaunaurd and Uberall

1. Introduction
1.1 The Beginnings
1.2 Understanding Sound
1.3 Representations of Sound
1.4 Sounds Classical and Sounds Quantum
1.5 Comparisons between Light and Ultrasound
1.6 The Adiabatic Idealization
1.7 Common Sense is Unsound
1.8 Scope of This Work
How to Use This Book

2. Water
2.1 Measurement of Sound Velocity
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2.1.1 Introduction
2.1.2 Accuracy and Errors
2.1.2.1 Temperature
2.1.2.2 Acoustical Delays
2.1.2.3 Impedance
2.1.2.4 The Control of Reverberation with Buffer Rods
2.1.2.5 Acoustical Bonds
2.1.2.6 Power Levels
2.1.2.7 Diffraction and Phase Cancellation
2.1.2.8 Timing Errors Due to Trigger Point Variation
2.1.2.9 Measuring Group Velocity
• 2.1.3 Calibration
2.2 The Dependence of Velocity of Sound on Density and
Compressibility
•
•
•
•
• 2.2.1 The Velocity of Sound in Mixtures and Suspensions
2.2.2 The Velocity of Sound in Air/Water Mixtures
2.2.3 The Importance of Removing Air from Samples
2.2.4 The Effects of Temperature on Propagation in Water
2.2.5 The Effects of Pressure on Propagation in Water
• 2.2.6 Sound Velocity in Equidensity Dispersions
2.3 The Relationship between Velocity and Attenuation —
Conditions of High Attenuation
2.4 The Compressibility of Solute Molecules
•
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2.4.1 Introduction
• 2.4.1.1 Empirical and Semiempirical Methods
• 2.4.1.2 Concentrations
2.4.2 Determining Partial Volumes
•
•
• 2.4.2.1 The Method of Intercepts
2.4.3 Apparent Molar Quantities
2.4.3.1 Apparent Specific Volume
2.4.3.2 Apparent Compressibility
• 2.4.3.3 Concentration Increments
2.4.4 The Dilute Limit
• 2.4.4.1 Partial Specific Volume and Partial Specific Adiabatic Compressibility
2.4.5 Sound Velocity and Concentration — The Urick equation
2.4.6 Determining the Compressibility of Solute Molecules — a
Summary
• 2.4.7 Experimental Data on Compressibility and Its Interpretation
Protein

3. MULTIPHASE MEDIA
3.1 Apparatus
3.2 Determining Composition in the Absence of Phase Changes
•
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• 3.2.1 Alcohol
3.2.2 Sugar
3.2.3 Concentration of a Dispersed Phase in a Colloidal Phase
3.2.4 Analysis of Edible Oils and Fats
3.2.5 Cell Suspensions
• 3.2.6 Temperature Scanning
3.3 Following Phase Transitions
•
•
• 3.3.1 General Comments
3.3.2 Attenuation Changes
3.3.3 Crystallizing Solids
• 3.3.4 Crystallization in Colloidal Systems.
3.4 Determination of Solid Fat Content
3.4.1 Introduction
3.4.2 General Method
•
•
3.4.2.1 Region I
3.4.2.2 Region III
• 3.4.2.3 Region II
3.4.3 Margarine
3.4.4 Chocolate
3.4.5 Accuracy
3.4.6 Anomalies Close to the Melting Point
3.4.7 Comparison with Dilatometry and pulsed Nuclear Magnetic
Resonance
3.4.8 Solid Content and Particle Size
3.5 Crystal Nucleation
•
• 3.5.1 Crystal Nucleation Rates
3.5.2 Ice
3.6 The Solution-Emulsion Transition and Emulsion Inversion
• 3.6.1 Emulsion Inversion
3.7 Determination of Emulsion Stability by Ultrasound Profiling
•
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•
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3.7.1 Introduction
3.7.2 History
3.7.3 The Leeds profiler
•
•
3.7.4 Interpretation of Ultrasound Velocity Profiles
3.7.4.1 Renormalization
3.7.4.2 Limits of Applicability of Renormalization Method
• 3.7.5 Examples of Profiling
Summary

4. SCATTERING OF SOUND
•
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4.1 Theories of Sound
4.2 A Comparison of Electromagnetic and Acoustic
Propagation
4.3 Scattering theory
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4.3.1 Why scattering theory?
4.3.2 What Is Scattering? Assumptions of Scattering Theory
4.3.2.1 Long Wavelength Limit
4.3.2.2 Low Attenuation
4.3.2.3 Plane Wave
4.3.2.4 Scattering Is Weak
4.3.2.5 Random Distribution of Particles
4.3.2.6 Adiabatic Approximation
4.3.2.7 Navier–Stoke’s Form for the Momentum Equation
4.3.2.8 Thermal Stresses Neglected
4.3.2.9 No Changes in Phase
4.3.2.10 Linearization of Equations
4.3.2.11 Temperature Variations
4.3.2.12 System Is Static
4.3.2.13 Particles Are Spherical
4.3.2.14 Infinite Time Irradiation
4.3.2.15 Pointlike Particles
4.3.2.16 No Overlap of Thermal and Shear Waves
4.3.2.17 No Interactions between Particles
•
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•
•
•
• 4.3.2.18 Lack of Self-consistency
4.3.3 A Description of Weak Scattering
4.3.3.1 Wave Potentials
4.3.3.2 Modes in a Pure Liquid
4.3.3.3 Thermoelastic Scattering
4.3.3.4 Viscoinertial Scattering
4.3.3.5 Scattered Waves Combine within the Transducer
4.3.4 Plane Wave Incident on a Single-particle
•
•
• 4.3.4.1 Introduction
4.3.4.2 Spherical Harmonics
4.3.4.3 Boundary Conditions
4.3.5 Scattering by Many Particles
• 4.3.5.1 Introduction
• 4.3.5.2 Multiple Scattering Theories
4.3.6 Numerical Calculations Using Scattering Theory.
• 4.3.6.1 Particle Size Distribution and Change in Phase
4.3.7 The Results of Scattering Theory
4.3.8 Simplified Scattering Coefficients
4.3.9 Working Equations
•
• 4.3.9.1 The Urick equation
4.3.9.2 The Multiple Scattering Result
•
• 4.3.9.3 The Modified Urick equation
4.3.9.4 Experimental Determination of the Scattering Coefficients
4.3.10 Multiple Dispersed Phases
4.3.11 MathCad Calculation Results
4.3.12 Experimental Validation of Acoustic Scattering
Theory
Scattering from Bubbles

5. ADVANCED TECHNIQUES
5.1 Particle Sizing.
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5.1.1 Introduction
5.1.2 Review
5.1.3 Theoretical Limitations of Acoustic Particle Sizing
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5.1.4 Relaxation Effects
5.1.5 Ultrasonic Methods of Particle Sizing
5.1.5.1 Simultaneous Measurement of Velocity and Attenuation
5.1.5.2 Determinination of Particle Size from Velocity and Attenuation
5.1.5.3 Bandwidth and Signal-to-Noise Ratio
5.1.5.4 A Particle Sizing Apparatus — Pulsed Method
5.1.5.5 Continuous-Wave Interferometer
5.1.5.6 Commercial Particle Sizing Apparatus
•
•
5.1.5.7 Electroacoustics
5.1.5.8 The Future— Measurement Systems
5.2 Propagation in Viscoelastic Materials
•
• 5.2.1 Introduction
5.2.2 Measuring Aggregation in Viscoelastic Materials
•
•
5.2.2.1 Introduction
5.2.2.2 Detecting Aggregation with Ultrasound Profiling
•
•
5.2.2.3 Computer Modeling
5.2.2.4 Aggregation of Casein
5.2.3 Frequency-Dependent Ultrasound Profiling •
• 5.2.4 Particle Size Effects in Ultrasound Profiling
5.3 Bubbles and Foams
5.4 Automation and Computer Tools
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5.4.1 The Computer as Controller
5.4.2 Windows
5.4.3 Prototyping
5.4.4 RS232C
5.4.5 IEEE Bus
5.4.6 Instrument Programming
5.4.7 Oscilloscope
• 5.4.7.1 Fourier Analysis
5.4.8 Timer–Counter
5.4.9 The UVM
5.4.10 Transducer Excitation
5.4.11 Cabling
5.4.12 Calibration
5.4.13 Sample Changer
5.4.14 Temperature Control
• 5.4.15 Data Storage and Analysis
Conclusion
APPENDIX, GLOSSARY, AND BIBLIOGRAPHY
Appendix A Basic Theory
Appendix B MathCad Solutions of the Explicit Scattering
Expressions
Glossary
Bibliography