The monitoring of the material properties of complex liquids and polymers used across industries has become very relevant because a variety of companies are increasingly utilizing these materials in their day-to-day production activities and this would help to avoid mishaps or accidents. A variety of ways exist that help in the monitoring of these properties but of particular interest to this work are the Ultrasonic and Dielectric methods.
This work proposes real-time simultaneous measurements of complex liquids and polymer materials using both Ultrasonic and Dielectric Spectroscopy. It tries to answer the question 'Will having real-time simultaneous measurements using the two different techniques be feasible and helpful for industry use? There are currently sensors used in industries which employ either of the single technologies and a critical evaluation of combining both technologies is presented in this work.
The processing of complex liquids and the mechanical integrity, health and safety of complex engineering structures need to be constantly monitored so as to identify problems such as fatigue and damage that might occur. Continuous process monitoring in gaseous, liquid or molten media is a fundamental requirement for process control (Hauptmann et al, 2002) Â and ultrasonic sensor systems have been used to significantly detect and enhance material properties. Dielectric sensors have also been used for process control in industries. Ultrasonic Sensors utilize ultrasonic transducers which convert electrical signals into ultrasonic waves and vice versa (Hauptmann et al, 1998), while the Dielectric sensors -----. Â Â
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Correlations have been made between ultrasonic parameters, such as velocity, attenuation and backscattering, and material properties controlling the mechanical behaviour and the reason for these correlations is that Â the same microstructural features which affect the ultrasonic measurement such as grain size, grain boundaries and inclusions also play an important role in determining material properties of interest (Thompson, 1996 via Achenbach, 2000).
AIMS AND OBJECTIVES
RELEVANCE TO INDUSTRIES
2.0 BACKGROUND OF STUDY
Introduction and what is expected in the chapter
2.1 NON DESTRUCTIVE EVALUATION (NDE)
Non-Destructive Evaluation (NDE), also called Non-Destructive Testing (NDT) is a general term used for any assessment, test or evaluation carried out on materials and structures to ensure it is fit for service without causing any damage to them in any way. The American Society of Non-destructive testing states that NDE is the examination of an object with technology that does not affect the object's future usefulness (Shull, Introduction to NDE). It has become the leading product testing standard.
Non-destructive evaluation has become an integral part of virtually every process in industry where product failure can result in accidents or bodily injury (Hellier, 2001).
There are a wide range of NDT techniques available in industries which include:
- Visual/Optical testing
- Eddy Current Testing
- Acoustic Emission
- Ultrasonic NDE
This is currently the most commercially successful NDE method because it is relatively cheap to implement. Non-destructive Evaluation (NDE) has a variety of applications including flaw detection, dimensional measurements and determining material properties. It requires the test component to maintain its original state prior any test procedure and is well used in engineering and quality assurance for product continuity and to satisfy reliability standards (Tong, 2001). Products can also be tested without interrupting the manufacturing process. Non-destructive Evaluation and Non-destructive Testing (NDT) are often used interchangeably but the former defines a more quantitative approach.
Talk about who uses NDE currently in the industry?
Legislation of standards on testing e.g. British standards
Number of companies that produce and sell Ult/Diel kits already
- Dielectric NDE
Ultrasonic technology can be traced back to work done by a french physicist, Pierre Curie in 1880 on the piezoelectric effect. Pierre and his brother Jacques discovered that asymmetrical crystals such as quartz and Rochelle salt (potassium sodium tartrate) generate an electric charge when mechanical pressure is applied (Angelfire, 2010). In 1881, they discovered that the same crystals were deformed and mechanical vibrations were generated on application of an electrical field.
One of the first uses of Ultrasonics was medical imaging diagnosis and this dates back to the 1930s and 1940s. In 1937, Karl Theodore Dussik, a psychiatrist and neurologist began studying ultrasonography in conjunction with his brother Friederich, a physicist. They attempted to use ultrasound to diagnose brain tumors but unfortunately, it was later determined by Guttner, in 1952, that the images produced by the Dussiks were variations in bone thickness and not of brain tumours (Newman et al, 1998). Consequently, the United States Atomic Energy Commission reported that ultrasound had no role in the diagnosis of brain tumors. There is also Douglass Howry, another pioneer from the 1940s, who concentrated more on the development of ultrasonic equipment and the applied theory of ultrasound rather than its clinical applications. However, it was not until the 1970s that the work of these and other pioneers came into fruition.
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Testing of non-human materials can be done either destructively or non-destructively. Ultrasonics has evolved with advancements in technology whereby testing can be done while also retaining the original structure of the test object prior to testing. This is referred to as Non-destructive Evaluation. Some of the advantages of NDE over destructive tests for industries include its ability to be used in process monitoring like chemical reactions and crystallizations (Unwin, 2010) and its inexpensive nature in terms of equipment costs and time. Presently, there are various applications of Ultrasonics NDE to various industries among which are defect detection in materials, dimensional measurements and characterization of materials. However, for the purpose of our study, more emphasis will be on Ultrasonic NDE as a means of studying characterization of complex liquids and polymer materials.
Ultrasonics is a branch of Acoustics (Ensminger,1988) that involves the application of high-pitched sound energy to materials, usually at frequencies beyond the human audible range between 20kHz and 25MHz. The mechanical vibrations/ultrasound signals then propagate through the material at a velocity dependent on the properties of the material. The signal will also undergo some form of attenuation which is also specific to the material properties. If a discontinuity exists in a structure (for example, where two material types are in contact) then reflections or echoes may occur.
It differs from other forms of energy because it cannot travel through a vacuum; it requires a medium to propagate. Sound energy travelling at higher frequencies have shorter wavelengths, allowing them to reflect from objects more readily (Angelfire, 2010). The received data is also much easier to analyse because of their high resolution. However, they are difficult to generate and measure.
A discontinuity present in the path of the wave would cause a portion of the incident energy from the transducer is reflected from the surface of the defect and possibly scattered as additional mechanical waves throughout the object. These reflections are picked up by the same transducer in a pulse-echo set-up or a second receiving transducer in a through-transmission set-up (Schmerr,1998).
The data received are represented in the following ways for analysis:
A typical Ultrasonic setup is basically made up of the following functional units:
(SCHEMATIC DIAGRAM OF A TYPICAL ULTRASONIC SET-UP)
- The Pulser/Receiver: It produces short duration electrical pulses and receives the echoes in form of electrical energy. The receiver part of the pulser/receiver also has an amplifier that amplifies the received electrical signals as they are very small in size.
- The Transducer: is used for excitation and usually contains a piezoelectric crystal which transforms the electric signals produced by the pulser into high frequency ultrasonic energy that propagates through the material in form of mechanical acoustic waves of particular wavelengths dependent on the material characteristics of the object being tested. It also transforms the reflected echo back into electric signals which are picked up by the receiver and shown on a display as A-scans.
- The Digitizer: This transforms the analog signals which are represented as a voltage versus time trace on the oscilloscope into digital signals for further signal processing and analysis by the computer (Schmerr, 1998).
- The Oscilloscope: This provides a display of the signal as a voltage versus time graph.
Single technique Ultrasonic sensors have been in use in industry since ...
Propagation of waves
The basic principles of waves and wave propagation are significant to this study as sensing will be done with the inner conductor acting as a waveguide. Sound travels as mechanical waves in solids, liquids and gases in the form of vibrations and oscillations of atoms about a mean position. The direction of propagation of these sound waves is relative to the displacement or oscillation of the material particle vibrations, causing various wave modes to exist. These include:
Surface Waves: Classic surface wave propagation example includes surface waves, Lamb waves, and Stonely waves (Rose, 2002).
Ultrasonic NDT guided wave propagation differs from Bulk wave propagation in the sense that it requires a boundary to propagate. Propagating waves along the boundaries of thin plates result in reflections,mode conversions and some superimposing with areas of constructive and destructive interferences that finally result in nicely behaved guided wave packets that can travel in the structure (Rose, 2002).
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Acoustic properties of waves
Waveguide; Acoustic properties, attenuation, velocity, material, dimensions, shape, propagation xtics, modes etcâ€¦
The velocity, wavelength and frequency of sound in an elastic material are related by the following equation:
v = Â Î».f Â Â Â (1.1)
v = velocity (m/s)
Î»= wavelength (m)
f= frequency (Hz)
Acoustic Impedance (Z)
Transmission and Reflection coefficients:
Ultrasonic Transducer Characteristics:
Applications of ultrasounds
Measurement of the viscosity of fluids
Cure monitoring process of epoxy
Determining the composition of materials
Measurement of the viscosity of fluids
Viscosity is a measure of the forces (shear stress or tensile stress) that oppose the relative motion of an object through a fluid. It can also be defined as the resistance per unit area of a fluid to deformation or to flow . The flow properties of fluids can be divided into three main groups (I) Newtonian; (ii) Non-Newtonian, time dependent (iii) Non-Newtonian, time independent. The viscosity of a Newtonian fluid is constant and independent of the applied shear rate (shear stress) , therefore does not increase or decrease with an increase or decrease in shear rate. However viscosity of a Newtonian liquid depends on temperature and pressure (and also the chemical composition of the fluid if the fluid is not a pure substance).
The relation between the applied shear rate and the obtained shear stress is constant over the whole shear rate range . Liquids which show Newtonian flow behaviour are often simple, single-phase liquids and solutions of liquids with low molecular weights . Example of this type of liquid is water.
In both non-Newtonian time dependent and non-Newtonian time independent, the relationship between the shear stress and the strain stress is not linear. Therefore a constant coefficient of viscosity cannot be defined (not constant). In Non-Newtonian time dependent liquid viscosity increases or decreases with the duration of the applied shear stress examples of these types of liquids are paints and yoghurt. While in a Non-Newtonian, time independent f viscosity either increases or decreases with increased stress. Examples of these types of liquid are suspensions of corn starch or sand in water, Paper pulp in water, latex paint, ice, blood, syrup, and molasses.
There are many various techniques for measuring viscosity such as Coaxial-Cylinder method, glass viscometer, piezoelectric response voltage and ultrasonics, each suitable to specific circumstances and materials. The selection of the right viscometer from the scores of instruments available to meet the need of any application is a difficult proposition , due to the complexity of the application and in some cases the limitations of the proposed instrument. Ultrasound techniques are the most widely applied in the study of liquid states (most complicated amongst the three states of matter) due to its simplicity and accuracy. In a journal titled "Comparison between different methods for viscosity Measurements" by M. Mekawy, H. Afifi and Kh. El-Nagar concluded that the viscosity measured by capillary glass, Coaxial-Cylinder rheometer ultrasonic pulse echo method showed a good correlation and can be applied industrially. This will add the calibration of viscometers in industry. From the ultrasonic results obtained, it is of interest to conclude that the ultrasonic parameters are very suitable as characterizing parameters for oils in the range mentioned.
Ultrasound has been used to measure the viscosity of fluids as early as 1949 . However prior to World War II (1935-1945), sonar, the technique of sending sound waves through water and observing the returning echoes to characterize submerged objects, inspired early ultrasound investigators to explore ways to apply the concept to medical diagnosis.
In 1929 and 1935, Sokolov studied the use of ultrasonic waves in detecting metal objects. Mulhauser, in 1931by using two transducers to detect flaws in solids. Firestone (1940) and Simons (1945) developed pulsed ultrasonic testing using a pulse-echo technique . Ultrasonic velocity and absorption measurements have proved to be useful in dealing with the problems of structure of liquids and interaction between the molecules. Ultrasonic pulse-echo method is in wide spread use for these measurements .
Ultrasonic velocity in a liquid system is conditioned by the state of molecular properties of the system [6-9]. In addition to temperature, the ultrasonic velocity in a liquid system is responsive to viscosity as well .
Measuring the ultrasonic viscosity
Pulse echo is the most appropriate method of measuring viscosity when using ultrasonic techniques. In pulse echo method, a short sinusoidal electrical wave activates the ultrasonic transducer. The transducer then produces sound wave train into the liquid inside the cell . This cell is made up of stainless steel, which avoids any chemical reactions between the chemicals and the cell. Some advantages of this method are that the sound velocity can be measured at the same time as the attenuation.
Polarized shear waves may be used as well as longitudinal waves, and a wide range of sound frequency may be employed.
Using an oscilloscope (60 MHz time base oscilloscope, Philips,Netherlands) direct measurement of the time required for the pulses to travel twice the length of the specimen is possible, which allows immediate calculation of the ultrasonic wave velocity as given in the following equation:
Where L is the liquid length and Î”t is the time interval. The velocity measurements using this method were carried out at a nominal frequency of 4 MHz (central frequency of 0.7 MHz and band width of 1.4 MHz) at temperature of 25oC. The estimated accuracy of the velocity measurement is about 0.6 %.
The ultrasonic attenuation and ultrasonic viscosity are given in the form;
Where An / An+1 is the ratio between two successive echoes An, An+1, Ï is the density of the oil and f is the ultrasonic frequency. The uncertainty of the measurements of ultrasonic attenuation and viscosity are Â±0.01 dB/cm and Â±0.03 mPa.s respectively.
The experiment was performed over ten oil samples at different densities and single temperature 25oC. Each experiment was repeated three times and through three days consecutively, and the median was chosen as an end result.
Viscosity is a principal parameter when any flow measurements of fluids, such as liquids, semi-solids, gases and even solids are made. Viscosity measurements are made in conjunction with product quality and efficiency. In industries, the knowledge of a material's viscosity behaviour and rheological characteristics gives manufacturers an important product dimension. It is essential in predicting pumpability and pourability factors, performance in dipping or coating operation, or the ease with which it may be handled, processed and used. The interrelation between rheology and other product dimensions often makes the measurement of viscosity the most sensitive or convenient way of detecting changes in color, density, stability, solids content, and molecular weight .
Many manufacturers now regard viscometers as a crucial part of their research, development, and process control programs. Viscosity measurements are often the quickest, most accurate and most reliable way to analyze some of the most important factors affecting product performance.
There are many different techniques for measuring viscosity, each suitable to specific circumstances and materials. The selection of the right viscometer from the scores of instruments available to meet the need of any application is a difficult proposition. Today's instruments vary from the simple to the complex: from counting the seconds for a liquid to drain off a stick to very sophisticated automatic recording and controlling equipment. This places the instrument user in a position in which his own appreciation of the flow phenomena involved, coupled with the instrument manufacturer's "know how and experience", must be brought to bear.
Experimental Set up (Pulse Echo)
Generator ( what are the ideal characteristics, signal types, signal types i.e. tone burst, sine burst
Examples of kit?
Prices of kit.materials
Transducers ( give a basic definition of transducers, types, sources (do we buy or make ourselves?)
Amplifier + receiver ( an amplifier is needed because the received signal would be small)
How would you set it up?
-Pulse echo systems
-Through-transmisison ( No need to mention angles as it's not relevant to our design)
- Experimental procedure
-What does the data look like? Data analysis and process?
Experimental set-up for pulse echo
Pulse Echo Mode
The general set-up for a pulse-echo technique is shown in fig.1. At low frequencies, it is convenient to excite longitudinal and torsional waves by means of magnetostrictive transduction. A section of the magnetostrictive waveguide is permanently magnetised . A coil, acting both as a transmitter and receiver (transducer) is placed at one end of the magnetised section and an alternating current is applied . This provides an additional time varying axial magnetic field . Longitudinal stress waves (modes) are created in the wire if it is axially magnetised by the time varying field inside the coil (joule effect); if it is circumferentially magnetised, torsional stress waves (modes) are excited (Wiedemann effect) .
2.3 DIELECTRIC SPECTROSCOPY
Dielectric Spectroscopy is the measurement of the dielectric properties of a material as a function of the frequency of an applied alternating electric field. It is a technique used to study the dynamics of a material when subjected to an electromagnetic field. The dynamics of the material is related to its polarisability under the effect of an applied electric field. Different materials characterized by different structural and molecular properties exhibit different types of polarisation which contribute to the overall polarisation in the dielectric material. The polarisation mechanisms determine the variation of the complex permittivity of the dielectric material with frequency. The complex permittivity of the test medium is related to the capacitance and conductance by its real and imaginary components respectively. A unique characteristic of dielectric spectroscopy is its use with a wide range of frequencies from 10-5Hz to 1016Hz and this enables measurement of dielectric response to both slow (low frequency) and fast (high frequency) molecular events [11,b].
The term "dielectric" was coined byÂ William WhewellÂ (from "dia-electric") in response to a request fromÂ Michael Faraday. The word "dia" is a Greek word meaning to cut across or block and "electric" implies displaying properties of electronic conduction. Thus dielectrics are often thought of crudely as non-conductors or "insulators". However, Von Hippel (a widely respected "dielectrician"), gave a more precise definition of dielectrics as not a narrow class of so-called insulators, but the broad expanse of non-metals considered from the standpoint of their interaction with electric, magnetic, of electromagnetic fields. Thus, this definition gave a better and broader understanding of dielectrics in the science community because it implies that dielectrics concerns gases as well as with liquids and solids, and also the storage of electric and magnetic energy as well as its dissipation.
Dielectric Spectroscopy goes back to the end of the nineteenth century. Majority of the contributions on dielectric spectroscopy can be attributed to Dutch-born U.S. physical chemist, Prof. Peter Joseph Wilhelm Debye. This is majorly due to his extensive work and his first important research, onÂ electric dipoleÂ moments. Polarisation can be defined to be the vector summation of all individual dipole moments in a material and since Polarisation and the dynamics of electric charges are at the heart of dielectrics, and are often described in terms of macroscopic properties such as permittivity, dielectric loss and breakdown strength, Prof. Debye's work plays an important role in understanding Dielectric Spectroscopy.
Over the years, after the appearance of the original Debye contributions between 1912 and 1913, on the subject, there had been an increasing interest in the field of Dipole moment and especially its contribution to the study of dielectric properties. Beginning in 1955 at least three monographs that summarize advances in the subject included: C. P. Smyth, "Dielectric Behavior and Structure," United States; N. E. Hill, W. E. Vaughan, A. H. Price, and M. Davies, "Dielectric Properties and Molecular Behavior," Great Britain; and V. I. Minkin, O. A. Osipov, and Yu. A. Zhdanov, "Dipole Moments in Organic Chemistry," the Soviet Union.
As already noted, polarization plays an important role in the study of dielectrics. There are four mechanisms of polarization namely: Electronic, Ionic Interfacial and Orientation Polarization. These mechanisms are described in detail later in this report but of major importance to Dielectric Spectroscopy is the Orientation Polarization. The concept of the orientation of dipolar molecules in an alternating electric field was applied by Debye (1913) in the explanation of the behaviour of the real and imaginary parts of two dielectric constants: Permittivity and dielectric loss factor which are better terms when frequency dependence is involved.
The basic principle of Orientation Polarization is that when the field is applied, or released, a finite time will be required for the molecules to come to their equilibrium orientation because there is a viscous resistance to these rotatory motions. The range of frequency over which the real dielectric constant is variable extends from the static field to one that oscillates so rapidly as not to provide for any rotational motion of the polar molecule at all; the theory thus describes a typical molecular relaxation process. The accompanying constant, called the time of relaxation, is made available from measurements of the frequency variation of either the real dielectric constant or the energy absorption for the system; in solutions this time constant may be related to molecular size and shape and thus, it's quite useful in cure monitoring of Polymers. The arguments and equations presented in connection with the frequency dependence problem have been of greatest interest in electrical engineering.
Early applications of Dielectric Spectroscopy in the studies of Polymers date back as far as 1951 when Debye along with F. Bueche, applied the relatively simple idea of internal rotations to an organic high-polymer system. Although, it is necessary to note that one may also gain information about the average size of coiled polymer molecules in solution from light scattering measurement. The dielectric properties of materials have come to play a fundamental role in the description of physical phenomena in many branches of modern science, technology, and engineering. Over the last several decades, the equivalent frequency range of Dielectric Spectroscopy (DS) has been expanded by various experimental techniques so dielectric relaxation processes of systems can be measured over an extremely wide range of characteristic times (10Â 5Â s - 10Â -12Â s).
As a result, these techniques now occupy a special place among numerous modern methods used in physical and chemical analyses of materials. Since dielectric spectroscopy measures the time evolution of molecular polarization, the technique is especially sensitive to intermolecular interactions and their role in molecular cooperative processes. Dielectric spectroscopy provides a link between the dynamics of molecular motion of the individual constituents of the complex material and the characterization of its bulk properties. The recent successful developments of the Time Domain Dielectric spectroscopy methods and Broadband Dielectric Spectroscopy have radically changed the role of dielectric spectroscopy as an effective tool for structural investigation in solids and liquids on macroscopic, microscopic and mesoscopic levels.
A dielectric material is an insulator -with a high characteristic resistivity- in which electrostatic dipoles exist either permanently or as a result of an applied electric field. Dielectrics act as non-conducting bridges between two conducting plates (in most cases metals). Depending on design specifications, the arrangement of the conducting plates can follow several capacitor models such as parallel plate and cylindrical capacitor models (shown in figure 1.1). A very fundamental property of dielectrics is their high polarisability. Electric polarisability is the relative tendency of a charge distribution, like the electron cloud of an atom or molecule, to be distorted from its normal shape by an external electric field, which may be caused by the presence of a nearby ion or dipole . Permittivity is a characteristic of dielectric materials and it is related to polarisation and response to an applied electric field.
1.2.1 DIELECTRIC PROPERTIES OF MATERIALS
1.2.2 DIELECTRIC IN CAPACITORS
Polarisation is the resultant effect of applying an electric field to a dielectric material. Dielectric materials basically consist of positively charged nucleus and a cloud of uniformly distributed electrons around the nucleus. When an electric field is applied, there is displacement of charges in the direction of the electric field (positive charges) and in an opposite direction (negative charges). The effect of the applied electric field (E) and resultant displacement is an induced dipole moment (m) defined by
m = qr
r Â is the displacement between the charges;
m and r are vector quantities.
The dipole moment's direction points towards the positive charge as shown in figure 1
Figure 1. Displacement of charges due to applied electric field.
The magnitude of the induced dipole moment is related to an atom or particle and these individual dipole moments are summed up for the whole volume of the dielectric material. This vector summation is the polarisation (P) of the dielectric material and is defined by
P = m
1.3.1 TYPES OF POLARISATION
The polarisation mechanisms that contribute to the permittivity of the material vary for different materials. For polymeric materials, the major polarisation mechanism are polarisation due to charge migration (atomic polarisation) and polarisation due to orientation of permanent dipoles (orientation polarisation) [2,8,9]. In emulsions, gels, Electro-rheological fluids (ERF) and other heterogeneous systems, the major polarisation mechanisms are the Maxwell-Wagner (interfacial) polarisation and orientation polarisation [3-7]. There are four types of polarisation depending on the induction mechanism and these are electronic, ionic, orientation and inter-facial polarisation and are briefly described below.
22.214.171.124 Electronic polarisation
This type of polarisation is induced as a result of displacement of electrons from their equilibrium positions with respect to the nucleus of the atom and thus induces a dipole moment. It is sometimes referred to as atomic polarization and is extremely fast with resonant frequency in the ultraviolet or visible range of the electromagnetic spectrum.
126.96.36.199 Ionic polarisation
This occurs mainly in ionic crystals and it is related to the displacement of positive ions (cations) and negative ions (anions) under the influence of an electric field. These displacements are made with slower kinetics when compared with electronic polarisation. Its resonant frequency is in the infrared range of the electromagnetic spectrum.
188.8.131.52 Orientation polarisation
This type of polarisation occurs when an electric field is applied to materials that have permanent dipoles. The resultant effect is the alignment of the dipoles of the material in the direction of the electric field. Orientation polarisation varies inversely with absolute temperature and its dispersion has a wide frequency range depending on the material . The limiting frequency can be as high as 1011 Hz in the microwave spectrum.
184.108.40.206 Interfacial polarisation
This polarisation is related to heterogeneous systems and is also referred to as space-charge polarisation or Maxwell-Wagner polarisation. It is due to accumulation of charges at the interface (or inter-phase) between components in the heterogeneous dielectric when subjected to a low frequency electric field (10-1 to 102 Hz) [8,9].
1.4 FREQUENCY RESPONSE/ FREQUENCY DEPENDENCE
1.5 TEMPERATURE DEPENDENCE
1.6 EXPERIMENTAL METHOD
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Tian Hao. The Interfacial Polarization-Induced Electrorheological Effect. JOURNAL OF COLLOID AND INTERFACE SCIENCE 206, 240-246 (1998).
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F. Bordi, C. Cametti, T. Gili. Dielectric spectroscopy of erythrocyte cell suspensions.
A comparison between Looyenga and Maxwell-Wagner-Hanai effective medium theory formulations. Journal of Non-Crystalline Solids 305 (2002) 278-284
Koji Asami, Eugen Gheorghiu, Takeshi Yonezawa. Dielectric behavior of budding yeast in cell separation. Biochimica et Biophysica Acta 13811998.234-240
Juan Martinez-Vega. Dielectric Materials for electrical engineering. (2010), John Wiley & Sons, Inc.
Friedrich Kremer, Andreas Schönhals. Broadband dielectric spectroscopy. (2003)
Dielectric properties of materials available online at http://www.kayelaby.npl.co.uk/general_physics/2_6/2_6_5.html
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. Pandey D J, Dubey P G, Shukla P B and Dubey N S, J. Pure Appl. Ultrason. 15 (1), 33, 1993.
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