Investigation of the Bouncing Behavior of Two Rubber Balls
Experiments were performed with two rubber balls, one black and the other red. Apart from their color, there was no apparent difference. However, when they were allowed to fall onto a hard surface, the red ball bounced back to almost the same height from which it had been dropped while the black ball hardly bounced at all. How can this behavior be explained?
Introduction
A characteristic property of rubber balls is that they bounce back up when they are dropped onto a hard surface. The two balls investigated in this article were very different in this respect.
The red ball bounced up to about 86% of the height from which it had been dropped. In contrast, the black ball bounced up to only about 2% of the initial dropping height. Can this behavior be predicted from certain material properties and if so how can we measure these properties?
The bouncing behavior of a ball depends on its viscoelastic properties. The red ball is "elastic" – its kinetic energy on hitting a hard surface is largely stored as deformation energy, which is afterward available for bouncing.
In contrast, the deformation energy of the black ball is transformed mainly into heat. This energy is no longer available for bouncing so the black ball bounces to a much lower height (Figure 1).
The material parameter that describes this behavior is the modulus of elasticity (Young's modulus). This consists of two components: the storage modulus that describes the energy storage capacity and the loss modulus that characterizes the dissipative processes in the material. The ratio of the loss modulus and the storage modulus is called the loss factor or tan δ. The loss factors can be used to estimate the ratio of the energies transformed into heat in the two balls, Ered/diss and Eblack/diss , according to equation 1 [1].
Bouncing experiments with the two balls yielded a value of 0.14 for this ratio. This means that when the balls hit the surface, about seven times more energy is dissipated in the black ball than in the red ball.
DSC Measurements
The two balls were first measured by DSC using a METTLER TOLEDO DSC823e equipped with an FRS5 sensor and a liquid nitrogen cooling system. The results of the measurements are shown in Figure 2.
The black ball exhibited a glass transition at about –60 °C. The behavior of the red ball was very different. After a glass transition at –102 °C, part of the material crystallized at about –58 °C; the crystallites that resulted melted at about –22 °C. The two rubber balls obviously consisted of different materials. Their actual compositions are listed in Table 1.
The results show that it is very easy to differentiate between the two types of rubber by DSC. However, the DSC curves do not tell us anything about the bouncing behavior of balls. To get this information, we have to use DMA.
DMA Measurements
The DMA measurements were performed in the shear mode using a METTLER TOLEDO DMA/SDTA861e . The samples had a diameter of about 6 mm and were 2 mm thick. In the shear mode, the shear modulus, G, is measured and not the elastic modulus, E. For isotropic, incompressible materials, these two quantities are related by equation 2.
E = 3∙G (2)
The materials used for the two balls fulfill these conditions to a good approximation.
Figure 3 displays the results of heating measurements in the range –125 °C to +60 °C at a heating rate of 1 K/min and at frequencies of 1 and 100 Hz. The upper diagram shows the storage modulus curves and the lower diagram the loss factor (tan delta) curves.
Below –100 °C, the red ball is in the glassy state. The first pronounced step in the storage modulus and the corresponding peak in tan delta are due to the glass transition. The frequency dependence of the temperature range of the step in the storage modulus and the temperature of the corresponding peak in tan delta are characteristic for a glass transition.
After the glass transition, the storage modulus increases slightly due to cold crystallization in the material. The next step in the storage modulus at about –25 °C is a result of the melting of the crystallites. The temperatures at which these two processes occur are independent of the frequency of the DMA oscillation.
The storage modulus curve of the black ball exhibits merely a broad step whose position is once again frequency dependent. The tan delta curve shows an overlap of two frequency-dependent peaks, which correspond to the glass transitions of isobutylene and isoprene.
The loss factor (tan delta) is decisive (see equation 1) for assessing the bouncing behavior of the two balls. From the values of tan δ at 20 °C, it can be seen that at this temperature the loss factor is very frequency dependent (at least for the black ball). But the question is, what frequency corresponds to the bouncing experiment performed here?
To estimate this we must model the process of the ball hitting the hard surface. With simplified assumptions, we obtain a characteristic time τ given by equation 3.
for the interaction time of the ball with the surface. Here ρ is the density of the ball, d its diameter and E the modulus of elasticity. For details of this how this is derived see reference [1]. In our case, d is 3 cm, the density of the balls is about 0.94 g/cm3 , and E is about 10 MPa for the black ball and about 3 MPa for the red ball (these values were estimated from the shear modulus curves shown in Figure 3 for 100 Hz and 20 °C using equation 2.
This yields a value of τ of about 0.0009 s for the black ball and a value of 0.00167 s for the red ball. This corresponds to frequencies of about 1100 Hz and 600 Hz.
To assess the bouncing behavior correctly at room temperature, the loss factor has to be determined at these frequencies. The DMA/SDTA861e cannot however perform measurements at frequencies above 1000 Hz.
Information about the modulus at frequencies that lie outside the frequency range of the instrument (like here for the black ball) can however be obtained by constructing a so-called master curve. This is done by performing frequency scans at different temperatures.
The resulting frequency scan curves are shifted along the frequency axis so that the start and ends of the individual measurement curves join up as well as possible. The master curve that results describes the frequency dependence of the modulus in a frequency range that is many decades greater than that which can be directly measured by the instrument. The master curve can be constructed for any reference temperature. Further details about master curves can be found in reference [2].
Figure 4 displays the master curves for G', G'' and tan δ for the black ball at 20 °C. It also shows the measured frequency scan curve for the loss factor for the red ball at 20 °C in the range 0.1 Hz to 1000 Hz.
The figure shows that at room temperature the black ball is a glassy material above 109 Hz and that the rubbery plateau begins below about 10 Hz; the glass transition lies between the two. The loss factor shows an overlap of two peaks. They correspond to the glass transitions of isoprene and isobutylene.
Conclusions
The bouncing behavior of the two balls is different because they are made of different materials. The materials can be easily differentiated by means of DSC measurements through their glass transition temperatures and their crystallization and melting behavior.
DMA measurements, however, are needed to characterize the bouncing behavior of the two balls. Here, it is important that the measurement conditions (frequency and temperature) correspond to the actual conditions of the bouncing experiment.
At room temperature, it was estimated that the frequencies relevant for the bouncing behavior were about 1100 Hz for the black ball and about 600 Hz for the red ball.
The viscoelastic properties corresponding to these conditions were determined from master curves and from frequency scans. The observed bouncing behavior of the two balls also agrees quantitatively with the results from the DMA measurements.
Investigation of the Bouncing Behavior of Two Rubber Balls | Thermal Analysis Application No. UC 404 | Application published in METTLER TOLEDO Thermal Analysis UserCom 40