**2.1.1 Rigid wall containers**

Since acoustic waves reveal useful information on the characteristic of the material through which they travel, measurement of acoustical properties such as velocity, attenuation, and phase changes resulting mainly from wave reflections in the transfer media are often used as a tool for mechanical characterization of materials. Specifically, ultrasound has found a wide range of applications in the measurement of the viscosity of liquids. Mason et al. (1949) first introduced an ultrasonic technique to measure the viscosity of liquids. They used the reflection of a shear wave in the interface between a quartz crystal and the sample liquid. Since then many other ultrasonics related techniques have been developed to measure viscosity of liquids [Roth and Rich (1953), Hertz et al. (1990), and Sheen et al., (1996)]. However, acoustical techniques that use sonic frequency have been rather limited. The main reason for this has been the lack of practical approaches that can employ frequencies in the sonic range to study the rheology of liquids. Tabakayashi and Raichel (1998) tried to use sonic frequency waves for rheological characterization of liquids. They affixed a hydrophone and a speaker to both ends of a cylindrical tube and analyzed the effect of the liquid non-Newtonian behavior on the propagation of the sound waves. The approach used by these authors is similar to the well known impedance tube method commonly applied to gases contained in cylindrical tubes, known as waveguides, but their analysis did not include the effect of the tube boundaries on sound propagation which can be of importance. In that sense, the application of the impedance tube techniques to test liquids has been limited because the loss of wall rigidity, i.e. the tube boundary, which normally it does not occur in tubes filled with air or air waveguides. Thus, when the tube is filled with a liquid, the tube may become an elastic waveguide and the rigid wall approximation loses its validity. The key assumption of having rigid walls is related to the shape of the acoustic wave moving through the liquid. In order to have manageable equations to estimate the viscosity of the liquid from acoustic measurements using these systems it is important to generate planar standing waves from which acoustic parameters can be readily obtained. Mert et al. (2004) described a model and the experimental conditions under which the assumption of standing planar waves stands.

If a tube with rigid walls is considered the propagation of unidirectional plane sound waves though the liquid contained in the rigid tube can be described by the following equation (Kinsler et al., 2000):

$$\frac{\partial^2 p}{\partial t^2} - \mathbf{C}\_1^2 \frac{\partial^2 p}{\partial x^2} \tag{1}$$

where *p* is the acoustic pressure, *t* is time, *x* distance and *C*1 the speed of the sound in the testing liquid. The solution of Equation (1) can be expressed in terms of two harmonic waves travelling in opposite directions and whose composition gives place to the formation of standing waves (Kinsler et al., 2000):

$$p(\mathbf{x}, t) = p^{+} e^{\int\_{\cdot}^{\cdot} i \alpha t + \hat{k}(L - x) \int\_{\cdot} \cdot \mathbf{p}^{-}} + p^{-} e^{\int\_{\cdot} i \alpha t - \hat{k}(L - x) \int\_{\cdot}} \tag{2}$$

*p+* is the amplitude of the wave traveling in the direction *+x* whereas *p-* is the amplitude of the wave traveling in the direction *–x* and *i* is the imaginary number equal to 1 . The complex wave number 1 <sup>ˆ</sup> *k Ci* / includes the attenuation due to the viscosity of the liquid, is the frequency of the wave. The corresponding wave velocity can be obtained from integration of the pressure derivative respect to the distance *x*, an equation that is known as the Euler equation (Temkin, 1981) and calculated as <sup>1</sup> ( , ) 0 *<sup>p</sup> vxt dt x* ), which

yields:

300 Biomaterials – Physics and Chemistry

material characterization in air have been around for almost a century. The technique is described in basic acoustics text books such as Kinsler et al., (2000) and Temkin (1981). Impedance tube methods based on standing waves and the transfer function method have been accepted as standard methods by the American Society for Testing and Materials (ASTM 1990 and 1995), thus they will not be discussed in this chapter. Instead the chapter will focus on liquids, semiliquid and semisolid materials, many of them exhibiting viscoelastic properties, i.e. those properties that are more representative of biomaterials

The theoretical background that supports mechanical characterization of materials using vibration/acoustic based methods is mainly based on the characterization of acoustic waves propagating though the material. In that sense the analysis can be classified on the type of

The analysis of liquid samples can be further classified based on the type of container used to confine the testing liquid. One of more important aspects to consider in this classification is the rigidity of the container walls. Two cases are considered: containers with rigid walls

Since acoustic waves reveal useful information on the characteristic of the material through which they travel, measurement of acoustical properties such as velocity, attenuation, and phase changes resulting mainly from wave reflections in the transfer media are often used as a tool for mechanical characterization of materials. Specifically, ultrasound has found a wide range of applications in the measurement of the viscosity of liquids. Mason et al. (1949) first introduced an ultrasonic technique to measure the viscosity of liquids. They used the reflection of a shear wave in the interface between a quartz crystal and the sample liquid. Since then many other ultrasonics related techniques have been developed to measure viscosity of liquids [Roth and Rich (1953), Hertz et al. (1990), and Sheen et al., (1996)]. However, acoustical techniques that use sonic frequency have been rather limited. The main reason for this has been the lack of practical approaches that can employ frequencies in the sonic range to study the rheology of liquids. Tabakayashi and Raichel (1998) tried to use sonic frequency waves for rheological characterization of liquids. They affixed a hydrophone and a speaker to both ends of a cylindrical tube and analyzed the effect of the liquid non-Newtonian behavior on the propagation of the sound waves. The approach used by these authors is similar to the well known impedance tube method commonly applied to gases contained in cylindrical tubes, known as waveguides, but their analysis did not include the effect of the tube boundaries on sound propagation which can be of importance. In that sense, the application of the impedance tube techniques to test liquids has been limited because the loss of wall rigidity, i.e. the tube boundary, which normally it does not occur in tubes filled with air or air waveguides. Thus, when the tube is filled with a liquid, the tube may become an elastic waveguide and the rigid wall approximation loses its validity. The key assumption of having rigid walls is related to the shape of the acoustic

material being tested, i.e. liquid, viscoelastic semifluid, and viscoelastic semisolid.

behavior.

**2.1 Liquid materials** 

**2.1.1 Rigid wall containers** 

**2. Vibration fundamentals and analysis** 

and containers with deformable/flexible walls.

$$w(\mathbf{x}, t) = P^{+} e^{\int\_{\mathbf{x}}^{\cdot} \left[ iat + \hat{k}(L - x) \right]} - P^{-} e^{\int\_{\mathbf{x}}^{\cdot} \left[ iat - \hat{k}(L - x) \right]} \tag{3}$$

By applying the boundary conditions such that at *x* = 0 the fluid velocity is equal to the velocity of the piston that creates the wave, which is 0 (0, ) *i t v t ve* . The other boundary condition is derived from the practical situation of using an air space on the other end of the tube, which mathematical provides a pressure release condition at *x = L*, written as *p(L, t) = 0*. With these boundary conditions the coefficients *P+* and *P* in Equation 3 can be calculated as:

$$P^{+} = \frac{\rho\_0 \mathbf{C}\_1 \mathbf{v}\_0}{2 \cos \hat{k} \, L} \qquad \text{and} \qquad P^{-} = -\frac{\rho\_0 \mathbf{C}\_1 \mathbf{v}\_0}{2 \cos \hat{k} \, L} \tag{4}$$

where <sup>0</sup> is the density of the liquid, in repose, and *v0* the amplitude of the imposed wave. An analysis made by Temkin (1981) showed that two absorption mechanisms produce the attenuation of the sound energy. One is due to the attenuation effects produced by the dilatational motion of the liquid during the acoustic wave passage. The other term arises from the grazing/friction motion of the liquid on the wall of the tube. It can be shown that the attenuation due to wall effects is significantly larger than the attenuation due to the

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 303

Fig. 1. Schematic of the apparatus to measure viscosity of liquids inside a thick wall rigid tube.

Density *kg/m3* 

1 17 800 1246 2 96 971 1010 3 990 969 967 4 4900 963 943 5 10,050 980 -

Table 1. Physical properties of the liquids used to validate theory; properties are at 25oC.

Figure 2 illustrates measured magnitude of the mechanical impedances Abs (1/Za0) liquids 2, 3, and 4 (see Table 1). It is clearly shown in the figure the effect of the liquid viscosity on the measured impedance, in general the higher is the viscosity of the liquid the lower is the value of the magnitude of the mechanical impedance at the peak/resonance frequency. The resonance frequency (frequency where peaks in impedance are obtained) is also affected by the viscosity of the liquid and tend to decrease when the viscosity of the liquid increases, which would be indicating a decrease in the sound velocity though the liquid. The latter in

Intrinsic sound velocity *m/s* 

Further details of this experimental setting can be found elsewhere in Mert et al. (2004). It has been hypothesized that plots of absolute mobility as a function of frequency should exhibit peaks and valleys corresponding to the resonance and anti-resonance frequencies of the standing waves from which the rheological properties of the liquid, notably its viscosity, can be obtained. To validate the theory described above, liquids with known physical properties including viscosity, density and sound velocity though them (the latter measured by a pulse-echo ultrasound method) were tested. Table 1 list viscosity of the liquids (liquids

1 to 4) used for this test along with their relevant physical properties.

Liquid Viscosity

*mPa . s* 

dilatational motion of the liquid (Herzfeld and Litovitz, 1959). Thus, a simplified form can be used to obtain the complex wave number ( ˆ *k* ) and ultimately the attenuation ( ) from which the fluid viscosity can be extracted. The complex wave number is given by the following equation:

$$
\hat{k} = k\_1 - i\alpha \tag{5}
$$

where the real number *k1* is calculated as 1 1 *k C* and is the attenuation of the wave due

to the viscosity of the testing liquid.

For measurement purposes it is convenient to estimate the acoustic impedance as *p/v*, which from Equations (2) and (3) after application of the Fourier transform to the pressure and velocity variables yields:

$$Z\_{a0} = \frac{\tilde{p}}{\tilde{\mathbf{v}}} = \frac{i\,\rho\_0 a\nu}{\hat{k}} \tan \hat{k}L = \frac{i\,\rho\_0 a\nu}{\hat{k}} \frac{\sin \hat{k}L}{\cos \hat{k}L} \tag{6}$$

Where *p* and v are the Fourier transformed pressure and velocity respectively. The acoustic impedance is a complex number, as well as its inverse, which is known as mobility. Thus, the magnitude or absolute value of the mobility *Abs(1/Za0)* can be calculated. From values of the acoustic impedance or mobility the complex wave number ˆ *k* can be obtained as well as its real and imaginary components, from which the viscosity of the liquid and the intrinsic sound velocity in the liquid of interest can be estimated from the following equation (Temkin, 1981):

$$\hat{k} = k\_1 - \frac{\dot{\mathbf{i}}}{RC\_1} \left[ \left( \frac{o\mu}{2\rho} \right)^{0.5} + \left( \frac{o\nu}{2\mathcal{C}\_1^2} \right) \left[ \left( \frac{\mu}{\rho} \right)^{0.5} + \left( \frac{\mu}{3\rho} \right)^{0.5} \right] + \frac{o\nu^2}{4\mathcal{C}\_1^2} \left( \frac{\mu}{\rho} \right)^{0.5} \right] \tag{7}$$

Viscosities of the liquids can be also estimated from the Kirchhoff's equation, which has been derived for waveguides containing a gas and are given by the following equation (Kinsler et al., 2000):

$$a = \frac{1}{RC\_1} \sqrt{\frac{\mu o}{2\rho\_0}}\tag{8}$$

The parameter *R* in Equations 7 and 8 is obtained from to the magnitude of the mechanical impedance at the resonance frequency (see Figure 2).

From Equation (6) can be observed that plots of the acoustic impedance becomes a maximum when ˆ cos( ) *kL* is a minimum, i.e. when the frequencies are given by the relationship <sup>1</sup> <sup>1</sup> ( ) 2 2 *<sup>C</sup> f n <sup>L</sup>* , where *n* = 0,1,2,3,…., which is in terms of the length of liquid L and the velocity of sound in the liquid *C*1, can provide a location of those maxima. Those theoretical observations were experimentally validated by Mert et al. (2004) using an experimental setup as that described in Figure 1.

dilatational motion of the liquid (Herzfeld and Litovitz, 1959). Thus, a simplified form can

which the fluid viscosity can be extracted. The complex wave number is given by the

1

*C* and

0 0

v ˆ ˆˆ cos *<sup>a</sup> <sup>p</sup> i i kL <sup>Z</sup> kL*

Where *p* and v are the Fourier transformed pressure and velocity respectively. The acoustic impedance is a complex number, as well as its inverse, which is known as mobility. Thus, the magnitude or absolute value of the mobility *Abs(1/Za0)* can be calculated. From values of

its real and imaginary components, from which the viscosity of the liquid and the intrinsic sound velocity in the liquid of interest can be estimated from the following equation

> 1 2 2 1 1 1

 

*RC C C*

Viscosities of the liquids can be also estimated from the Kirchhoff's equation, which has been derived for waveguides containing a gas and are given by the following equation

> 1 *RC* 2

The parameter *R* in Equations 7 and 8 is obtained from to the magnitude of the mechanical

From Equation (6) can be observed that plots of the acoustic impedance becomes a maximum when ˆ cos( ) *kL* is a minimum, i.e. when the frequencies are given by the

and the velocity of sound in the liquid *C*1, can provide a location of those maxima. Those theoretical observations were experimentally validated by Mert et al. (2004) using an

2 3 2 4

1 0

, where *n* = 0,1,2,3,…., which is in terms of the length of liquid L

 

<sup>ˆ</sup> sin <sup>ˆ</sup> tan

 

0.5 0.5 0.5 0.5 2 2

 

*k k kL*

For measurement purposes it is convenient to estimate the acoustic impedance as *p/v*, which from Equations (2) and (3) after application of the Fourier transform to the pressure and

<sup>1</sup> <sup>ˆ</sup> *kk i*

*k*

*k* ) and ultimately the attenuation (

(6)

 

(8)

 

(5)

is the attenuation of the wave due

*k* can be obtained as well as

(7)

) from

be used to obtain the complex wave number ( ˆ

where the real number *k1* is calculated as 1

0

the acoustic impedance or mobility the complex wave number ˆ

to the viscosity of the testing liquid.

velocity variables yields:

(Temkin, 1981):

(Kinsler et al., 2000):

relationship <sup>1</sup> <sup>1</sup> ( ) 2 2 *<sup>C</sup> f n <sup>L</sup>*

ˆ

*<sup>i</sup> k k*

impedance at the resonance frequency (see Figure 2).

experimental setup as that described in Figure 1.

following equation:

Fig. 1. Schematic of the apparatus to measure viscosity of liquids inside a thick wall rigid tube.

Further details of this experimental setting can be found elsewhere in Mert et al. (2004). It has been hypothesized that plots of absolute mobility as a function of frequency should exhibit peaks and valleys corresponding to the resonance and anti-resonance frequencies of the standing waves from which the rheological properties of the liquid, notably its viscosity, can be obtained. To validate the theory described above, liquids with known physical properties including viscosity, density and sound velocity though them (the latter measured by a pulse-echo ultrasound method) were tested. Table 1 list viscosity of the liquids (liquids 1 to 4) used for this test along with their relevant physical properties.



Figure 2 illustrates measured magnitude of the mechanical impedances Abs (1/Za0) liquids 2, 3, and 4 (see Table 1). It is clearly shown in the figure the effect of the liquid viscosity on the measured impedance, in general the higher is the viscosity of the liquid the lower is the value of the magnitude of the mechanical impedance at the peak/resonance frequency. The resonance frequency (frequency where peaks in impedance are obtained) is also affected by the viscosity of the liquid and tend to decrease when the viscosity of the liquid increases, which would be indicating a decrease in the sound velocity though the liquid. The latter in

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 305

Extracted viscosities from impedance measurements along reported values of the liquid

Liquid viscosities were estimated from Equations 7 and 8 but better accuracy was obtained with Equation 7 that was derived for waveguides containing liquids (Mert et al., 2004).

When the walls of the waveguides are not rigid, during propagation of the waves the walls of the container expand and the overall liquid bulk modulus decreases leading to reduced sound velocity in the liquids. In addition, the fluid velocity cannot be assumed to be 0 at the can wall due to the expansion of the wall. These conditions make the governing equations used to estimate the wave attenuation and other relevant acoustic parameters very elusive, thus it is not possible estimate the liquid viscosity directly. One can overcome this problem by an empirical approach, which is by defining the quality factor *Q*, which is determined

2 1

<sup>2</sup> are the frequencies at which the amplitude of mechanical impedance response is

2000). If the quality *Q* is known in a given container-liquid system it would be possible to have an estimation of the viscosity of the liquids contained in the container, which would serve for quality control purposes. To prove that concept Mert and Campanella (2007) performed a study where a shaker applied vibrations to a cylindrical can containing a liquid using a system, schematically shown in Figure 1. The vibration was able to move the can containing the liquid, and a wave was generated through the liquid, which reflected back in the interface between the liquid and the headspace to form standing waves resulting from composition of the forward and reflected wave. Properties of the standing waves were measured in the frequency domain, and in particular the resonance frequency and the amplitude of the wave at that resonant frequency were obtained and using a calibration curve approach related empirically to the rheological properties of the liquid. It is important to note that these measurements do not provide the true viscosity of the testing liquid because the walls of the can are not rigid and deform significantly due to the vibration. Despite of that the properties of the standing waves, measured by the quality *Q*, were highly correlated with the rheology of the liquid, which was tested offline in a rheometer (Mert and Campanella, 2007). Results shown in Figure 4 are a frequency spectra for the liquids

Quality factors *Q* for the testing liquids can be estimated from the two peaks observed in Figure 4 and potted as a function of the liquid viscosity as illustrated in Figure 5. However, as shown in Figure 4, the first resonances peaks do not seem to provide sufficient resolution and the second resonance peaks, at higher frequency, are used to find a relationship with

Although interesting from an academic standpoint correlations like the one shown in the Figure 5 are not of practical applicability. However, if the can/cylinder contains a product

(9)

<sup>0</sup> is the resonance frequency (Kinsler et al.,

*<sup>o</sup> Q* 

viscosities are illustrated in Figure 3.

**2.1.2 Flexible wall containers** 

from the following equation:

reported in Table 1.

the liquid viscosity.

equal to half the actual value at resonance and

<sup>1</sup> and

Results of the calculated viscosities are illustrated in Figure 3.

fully agreement with the ultrasonic pulse-echo measurements used to estimate the sound velocities in the different liquids, which are reported in Table 1.

Fig. 2. Effect of viscosity on the magnitude of acoustical impedances measured in a setup as illustrated in Figure 1 at the driver position x = 0. Liquids 2, 3 and 4 were tested and properties and given in Table 1.

Fig. 3. Viscosities obtained from the different resonance frequencies for 4 different liquids obtained from the measured mechanical impedance and from Equation (7).

Extracted viscosities from impedance measurements along reported values of the liquid viscosities are illustrated in Figure 3.

Liquid viscosities were estimated from Equations 7 and 8 but better accuracy was obtained with Equation 7 that was derived for waveguides containing liquids (Mert et al., 2004). Results of the calculated viscosities are illustrated in Figure 3.

#### **2.1.2 Flexible wall containers**

304 Biomaterials – Physics and Chemistry

fully agreement with the ultrasonic pulse-echo measurements used to estimate the sound

 Sample 2 =96*mPa.s* Sample 3 =990 *mPa.s* Sample 4 =4900 *mPa.s*

Fig. 2. Effect of viscosity on the magnitude of acoustical impedances measured in a setup as illustrated in Figure 1 at the driver position x = 0. Liquids 2, 3 and 4 were tested and

0 1000 2000 3000 4000 5000

Frequency (Hz)

Fig. 3. Viscosities obtained from the different resonance frequencies for 4 different liquids

0 1000 2000 3000 4000 5000

Frequency (Hz)

 Measurement Liquid 1 Measurement Liquid 2 Measurement Liquid 3 Measurement Liquid 4 Reference Viscosity

obtained from the measured mechanical impedance and from Equation (7).

velocities in the different liquids, which are reported in Table 1.

properties and given in Table 1.

1

10

100

Viscosity (mPa.s)

1000

10000

0

1

2

3

Abs (1/Za0) x 104 (m/Pa.s)

4

5

When the walls of the waveguides are not rigid, during propagation of the waves the walls of the container expand and the overall liquid bulk modulus decreases leading to reduced sound velocity in the liquids. In addition, the fluid velocity cannot be assumed to be 0 at the can wall due to the expansion of the wall. These conditions make the governing equations used to estimate the wave attenuation and other relevant acoustic parameters very elusive, thus it is not possible estimate the liquid viscosity directly. One can overcome this problem by an empirical approach, which is by defining the quality factor *Q*, which is determined from the following equation:

$$Q = \frac{\alpha\_o}{\alpha\_2 - \alpha\_1} \tag{9}$$

<sup>1</sup> and <sup>2</sup> are the frequencies at which the amplitude of mechanical impedance response is equal to half the actual value at resonance and <sup>0</sup> is the resonance frequency (Kinsler et al., 2000). If the quality *Q* is known in a given container-liquid system it would be possible to have an estimation of the viscosity of the liquids contained in the container, which would serve for quality control purposes. To prove that concept Mert and Campanella (2007) performed a study where a shaker applied vibrations to a cylindrical can containing a liquid using a system, schematically shown in Figure 1. The vibration was able to move the can containing the liquid, and a wave was generated through the liquid, which reflected back in the interface between the liquid and the headspace to form standing waves resulting from composition of the forward and reflected wave. Properties of the standing waves were measured in the frequency domain, and in particular the resonance frequency and the amplitude of the wave at that resonant frequency were obtained and using a calibration curve approach related empirically to the rheological properties of the liquid. It is important to note that these measurements do not provide the true viscosity of the testing liquid because the walls of the can are not rigid and deform significantly due to the vibration. Despite of that the properties of the standing waves, measured by the quality *Q*, were highly correlated with the rheology of the liquid, which was tested offline in a rheometer (Mert and Campanella, 2007). Results shown in Figure 4 are a frequency spectra for the liquids reported in Table 1.

Quality factors *Q* for the testing liquids can be estimated from the two peaks observed in Figure 4 and potted as a function of the liquid viscosity as illustrated in Figure 5. However, as shown in Figure 4, the first resonances peaks do not seem to provide sufficient resolution and the second resonance peaks, at higher frequency, are used to find a relationship with the liquid viscosity.

Although interesting from an academic standpoint correlations like the one shown in the Figure 5 are not of practical applicability. However, if the can/cylinder contains a product

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 307

whose viscosity is an important quality parameter, the present method would be of great applicability because the viscosity of the liquid could be assessed quickly without the necessity of opening the can (i.e. non-destructive method). One example of such application is testing cans of tomato products, which are widely produced and consumed in most part of the world and the viscosity of these products is a key parameter associated to their quality in terms of sensory evaluation and processing applications. Viscosity of tomato products is evaluated using a number of rheological techniques that range from empirical to more fundamental methods. One standard method is the use of the BrookfieldTR viscometer. This viscometer is based on the rotation of a particular element (spindle) inside a can containing the product. The viscosity of the product is obtained basically from the resistance offered by the product to the rotation of the spindle provided shear rates and shear stresses can be accurately calculated. Since the geometries of the spindles often are not regular it is difficult to estimate the rheological parameters, i.e. shear stress and shear rate, which enable the calculation of the liquid viscosity. Tomato paste is a very viscous product that exhibits an important structure that can be destroyed during testing. One approach to overcome this problem is the use of a helical-path spindle, which through a helical movement continuously is touching a fresh sample. Given the complicated rheology, results of this test are considered empirical. The other approach is to use technically advanced rheological equipment, which though the use of geometries such as parallel plates the sample can be minimally disturbed. For these cases the true viscosity of these tomato purees can be obtained. These materials are known as non-Newtonian to indicate that their viscosities are a function of the shear rate. Most of the non-Newtonian liquids can be described by a rheological model known as the power-law model which can be expressed by the following

*n* 1

is the apparent viscosity of the material, which is a function of the applied shear

 and *k* and *n* are rheological parameter known as the consistency index and the flow behavior , respectively. The value of *k* is an indication of the product viscosity whereas *n* gives a relation between the dependence of the viscosity with the applied shear rates. For tomato products *n*<1, and the smaller is the value of *n* the largest is the effect of the shear rate on the viscosity of the liquid. To test the feasibility of vibration methods, cans of tomato puree and dilutions ranging from 23% to 3.5% solids were tested in an apparatus similar to the one shown in Figure 1. Results of the tests can be observed in Figure 6, where frequency spectra resulting from some the tests are shown. Figure 7 shows possible correlations between the quality *Q* measured from the frequency spectra data and the parameter *k*

*k* (10)

 

determined using a standard rheological technique and a parallel plate geometry.

As indicated in Figure 6 the frequency spectra peaks shift to lower frequencies when the solid content of the concentrates increases. In the range of frequency tested the two peaks are visible for concentrates with low solid content whereas the second peak at higher frequencies disappears for concentrates with higher higher solid contents (23 Brix). The amplitude of the Absolute value of the mobility is considerably decreased when the solid content and the viscosity of the concentrate increase (Figure 6). Given the existence of two peaks quality values can be extracted from the two peaks and relate them with the measured rheological properties, in this case the value of the consistency index (Figure 7). It

equation:

where

rate 

Fig. 4. Amplitudes at resonance frequencies for standard liquids given in Table 1. Curves were shifted up for clarity.

Fig. 5. Quality factor *Q* versus viscosity of liquid standards

 Liquid 1 = 17 mPa.s Liquid 2 = 96 mPa.s Liquid 3 = 990 mPa.s Liquid 4 = 4,900 mPa.s Liquid 5 = 10,050 mPa.s

Fig. 4. Amplitudes at resonance frequencies for standard liquids given in Table 1. Curves

0 500 1000 1500

Frequency (Hz)

Liquid 2

Liquid 1

10 100 1000 10000

Viscosity (mPa.s)

Liquid 5

Liquid 4

Liquid 3

Fig. 5. Quality factor *Q* versus viscosity of liquid standards

1

10

*Q*

100

0

20000

40000

60000

Abs (1/Za0) m/Pa.s

80000

100000

were shifted up for clarity.

whose viscosity is an important quality parameter, the present method would be of great applicability because the viscosity of the liquid could be assessed quickly without the necessity of opening the can (i.e. non-destructive method). One example of such application is testing cans of tomato products, which are widely produced and consumed in most part of the world and the viscosity of these products is a key parameter associated to their quality in terms of sensory evaluation and processing applications. Viscosity of tomato products is evaluated using a number of rheological techniques that range from empirical to more fundamental methods. One standard method is the use of the BrookfieldTR viscometer. This viscometer is based on the rotation of a particular element (spindle) inside a can containing the product. The viscosity of the product is obtained basically from the resistance offered by the product to the rotation of the spindle provided shear rates and shear stresses can be accurately calculated. Since the geometries of the spindles often are not regular it is difficult to estimate the rheological parameters, i.e. shear stress and shear rate, which enable the calculation of the liquid viscosity. Tomato paste is a very viscous product that exhibits an important structure that can be destroyed during testing. One approach to overcome this problem is the use of a helical-path spindle, which through a helical movement continuously is touching a fresh sample. Given the complicated rheology, results of this test are considered empirical. The other approach is to use technically advanced rheological equipment, which though the use of geometries such as parallel plates the sample can be minimally disturbed. For these cases the true viscosity of these tomato purees can be obtained. These materials are known as non-Newtonian to indicate that their viscosities are a function of the shear rate. Most of the non-Newtonian liquids can be described by a rheological model known as the power-law model which can be expressed by the following equation:

$$
\eta = k \dot{\gamma}^{n-1} \tag{10}
$$

where is the apparent viscosity of the material, which is a function of the applied shear rate and *k* and *n* are rheological parameter known as the consistency index and the flow behavior , respectively. The value of *k* is an indication of the product viscosity whereas *n* gives a relation between the dependence of the viscosity with the applied shear rates. For tomato products *n*<1, and the smaller is the value of *n* the largest is the effect of the shear rate on the viscosity of the liquid. To test the feasibility of vibration methods, cans of tomato puree and dilutions ranging from 23% to 3.5% solids were tested in an apparatus similar to the one shown in Figure 1. Results of the tests can be observed in Figure 6, where frequency spectra resulting from some the tests are shown. Figure 7 shows possible correlations between the quality *Q* measured from the frequency spectra data and the parameter *k* determined using a standard rheological technique and a parallel plate geometry.

As indicated in Figure 6 the frequency spectra peaks shift to lower frequencies when the solid content of the concentrates increases. In the range of frequency tested the two peaks are visible for concentrates with low solid content whereas the second peak at higher frequencies disappears for concentrates with higher higher solid contents (23 Brix). The amplitude of the Absolute value of the mobility is considerably decreased when the solid content and the viscosity of the concentrate increase (Figure 6). Given the existence of two peaks quality values can be extracted from the two peaks and relate them with the measured rheological properties, in this case the value of the consistency index (Figure 7). It

The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials 309

 First Peak Second Peak Linear Fit

Fig. 7. Quality factor versus the consistency index *k* value for tomato puree and its dilutions

1 10 100

*k* (Pa.s<sup>n</sup> )

Squeezing flow is a well-known technique that has been applied to characterize the properties of various biomaterials ranging from liquids to semisolids. The traditional method involves measuring the force required to squeeze a sample between two cylindrical disks either at a constant velocity or by applying a constant force or stress (Campanella and

The oscillatory squeezing low method (OSF) uses the same geometry as the standard squeezing flow method but it involves the application of small amplitude oscillations at random frequencies up to 20 kHz (Mert and Campanella, 2008). The method allows one to calculate both the viscous and elastic components of the sample viscoelasticity by measuring the response of the material in terms of force and acceleration to those oscillations. Transformation of the measured force and acceleration to the frequency domain yields a frequency spectrum for the sample and, ultimately, its resonance frequency. From analysis of this frequency response, two important viscoelastic properties of the samples, the loss

The application of acoustic principles to the squeezing flow method is a novel technique, which convert the squeezing flow method into OSF methos can measure the rheological properties of materials that range from pure liquids to solids. Advantages of this method include it being non-invasive, little to no sample preparation, and its ability to monitor rapid

A schematic of the OSF testing apparatus is illustrated in Figure 8. The design uses a piezoelectric crystal stack attached to an impedance head. Upon the application of voltage, the upper plate oscillates, and the force and acceleration at the oscillating plate are measured

modulus G" (viscous) and the storage modulus G' (elastic), can be obtained.

**2.2 Viscoelastic materials 2.2.1 Semifluid materials** 

1

10

*Q*

100

changes during dynamic processes.

Peleg 2002).

can be seen in the figure that correlations of the liquid rheological properties with its acoustic properties measured by the quality factor *Q* are strong and that the quality evaluated at the first peak provides a better representation of the liquid viscosity. This is also enhanced due to the presence of two peaks for low and moderate solid contents (measured as Brix), which provides more experimental points to establish the correlation (see Figure 7 – and compare first peak and second peak data).

Fig. 6. Frequency response spectra of tomato concentrates at different soluble solids concentrations, measured as Brix. Curves were shifted up for clarity.

Fig. 7. Quality factor versus the consistency index *k* value for tomato puree and its dilutions
