ABSTRACT

I. INTRODUCTION

K–12 science instruction is currently experiencing a revolution. Instruction is shifting from a view of standards-based instruction as a checklist of items to be covered to a three-dimensional approach wherein subject specific performance expectations are accomplished by the integration of three key components or dimensions (1). These dimensions are disciplinary core ideas (DCIs), crosscutting concepts (CCCs), and science and engineering practices (SEPs). These performance expectations are a part of the Next Generation Science Standards (NGSS) adopted by a majority of the United States as well as a number of other countries. As a part of a physics unit on waves, for example, students are expected to use mathematical equations to understand the relationship between wave characteristics, such as amplitude, wavelength, and frequency. This performance expectation targets a DCI of wave properties with the knowledge that wavelength and frequency are related to wave speed, which depends on the medium through which the wave is traveling. Supporting this DCI are ideas about cause and effect within CCCs and using mathematics and computational thinking within SEPs. Furthermore, these three dimensions are to be taught in an authentic and cohesive fashion.

As classroom teachers work to achieve this shift, 1 problem remains: the existence of a disconnect between what high school students and college researchers do. One reason for this disconnect is the immense cost that surrounds the highly technical equipment used in college and professional labs. The solution to this issue is the design and creation of an affordable and portable version of such materials that can be used by students to study advanced lab practices before graduating from high school. Students can observe phenomena relevant to engineering and even biomedical applications in the classroom. Here, we present a low-cost portable rheometer that can be used to measure the mechanical properties of many different materials. We characterize the stiffness of agar and other common classroom materials at frequencies that range from approximately 1 to 50 rad/s. This rheometer operates on the principles of damped harmonic motion, and we present a step-by-step guide for high school and undergraduate teachers interested in introducing mechanical concepts in the classroom.

II. BACKGROUND

In introductory-level physics and engineering courses, the mechanical properties of materials are often framed by 2 distinct categories: solids and fluids. In reality, however, most soft condensed materials, such as foams, gels, and biological fluids, exhibit a combination of both solid and fluid mechanical properties. These types of materials are called viscoelastic. The study of viscoelastic materials is an integral part of understanding structural stability in engineering, the importance of texture in food science, and the wear on joints in the body.

Whether a material is solid, fluid, or some combination thereof is often assessed by applying an external force on the material and measuring its resulting deformation. Purely elastic solids behave similarly to Hookean springs: the force F to stretch or compress the elastic solid is proportional to the amount of stretch or compression x, (1) $$F\text{ = }-kx$$ where k is the spring constant and the minus sign represents the restoring force opposing the displacement of the spring. The behavior of liquids can be modeled by Newton's law of viscosity, (2) $$F\text{ = }-bdx\text{/}dt$$ where b is a linear damping coefficient arising from the fluid viscosity and the damping effect is represented by the minus sign.

Viscoelastic materials can be described in part like solids and in part like fluids in relation to how they move in response to applied forces (2, 3). One such description is the Kelvin–Voigt model (Fig 1). The Kelvin–Voigt model captures viscoelastic materials as a combination of an elastic spring and a “dashpot,” a movable plunger immersed in a viscous fluid. In this model, the two elements are combined in parallel, as shown in Figure 1. That is, the two elements have the same displacement x, and the overall force on the material is the sum of the forces from each element, (3) $$F=-kx-bdx/dt$$

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Rotational oscillatory shear is one of the most common methods to quantify the viscous and the elastic components of a material (3) (Fig 2). When a strain is imposed on the sample, the sample resists the deformation by developing a restoring torque. Using the Kelvin–Voigt model, we have that the torque $\tau$ that the sample exerts on the upper plate is directly proportional to the twist angle $\theta$ and the twisting rate $d\theta /dt$ such that (4) $$\tau =-\kappa \theta -\beta d\theta /dt$$ where $\kappa$ and $\beta$ are angular analogs of and b and the minus sign again represents the forces opposing the angular displacement of the plate. Setting $\tau =I{d}^2\theta /d{t}^2$, where I is the moment of inertia of the plate, yields the equation (5) $$I{d}^2\theta /d{t}^2+\beta d\theta /dt+\kappa \theta =0.$$

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This is the equation of a damped harmonic oscillator. If the plate is given a twist at time $t=0$, then the family of solutions that starts with $\theta \left(0\right)=0$ is given by (6) $$\theta (t)={\theta }_0{e}^{-\beta t/2I}\sin \omega t$$ where ${\theta }_0$ is an amplitude that depends on the initial angular velocity of the disturbance. This describes the motion of the top plate (Fig 3), which dies off while oscillating at an angular frequency $\omega$ given by (7) $$\omega =\sqrt{\frac{\kappa }{I}-{(\frac{\beta }{2I})}^2}$$

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In this test, the elastic solid component of the material acts to oscillate the sample back and forth, while the fluid viscosity of the sample acts to dampen its motion. This motion is described by a slightly damped harmonic oscillator due to the spring and dashpot components, which represent the material's response (Kelvin–Voigt model).

Most commercial rheometers operate by applying a constant oscillatory stress or strain on the sample over time. As the sample is driven by the oscillations, the sample exerts against the upper plate a restoring torque that oscillates at the same frequency of the driven motion but at a phase lag depending on the amount of viscous damping in the sample. The resulting torque values and phase lag can be used to compute the sample's oscillatory elastic storage modulus G′ and viscous loss modulus G″ (see Supplemental Material).

Commercial rheometers rely on sensitive force transducers to measure the restoring torque the sample exerts on the plate as it is sheared and can cost between $40,000 and $200,000. Here, we present a low-cost portable torsion rheometer that can quantitatively measure G′ and G″ of soft gels and that can be assembled for less than $400. The torsion rheometer operates on the principles of damped harmonic motion, making it practical for educational purposes.

III. MATERIALS AND METHODS

A. Torsion rheometer assembly

Our torsion-based rheometer stems from an adopted version of the oscillatory test and is related to the torsion pendulum developed previously (4, 5). The design is shown in Figure 4. The torsion rheometer itself consists of a flat circular plate connected to a rotary motion sensor and a few hardware accessories to properly mount the plate above a sample. To allow precise real-time feedback of the plate motion, we designed the device around a rotary motion sensor (Vernier, Beaverton, OR) (Fig 4 and Table 1). The LabQuest 2 digital interface (Vernier) connects with the rotary sensor to display a graph of the rotary motion data while the data are being collected, but a computer may also be connected to the LabQuest to collect the angular position $\theta$ and angular displacement rate $d\theta /dt$ of the plate over time t from the Vernier rotary sensor.

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Table 1 Main parts for torsion rheometer.

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A building plan and a full list of the parts to assemble the torsion pendulum are provided in the Supplemental Material. The torsion rheometer requires a smooth top plate attached to the rotary sensor. While nearly all the hardware to assemble the torsional rheometer is commercially available, the top plate should be customized to interface with the rotary sensor (Supplemental Material). The Vernier sensor is then attached to a vertical pole mounted on an optical breadboard and then lowered to sandwich the test sample between the sensor's rotating plate and a flat surface, such as a glass plate.

For use with soft, slippy samples like agar, sandpaper can be added to the top and bottom plates with double-sided tape to grip the material and suppress slip between the material and the plates during oscillations.

B. Test materials and sample preparation

The benchtop rheometer was tested on 4 soft materials: polydimethylsiloxane (PDMS; Dow Corning, Midland, MI), JELL-O Jigglers (Kraft Foods, Chicago, IL), agar (BD Bacto, Fisher Scientific, Waltham, MA), super firm tofu (Trader Joe's, Monrovia, CA), and polyacrylamide (PAA) gels (Fisher Scientific, Waltham, MA). PDMS samples were prepared by mixing the 2-component dielectric gel at a ratio of 1:1, degassing the sample, and curing the sample at 65° for 1 h. The JELL-O Jigglers are a stiffer gelatin recipe with a higher gelatin-to-water ratio compared with conventional JELL-O recipes. JELL-O Jigglers were prepared by mixing 1 3-ounce package of JELL-O gelatin with 1.5 cups of boiling water as per the manufacturer's instructions. Agar was prepared by dissolving 2% agar by weight in water and bringing to a boil; 15 mL of agar was then poured into 100-mm Petri dishes and allowed to cool. The PAA gels were prepared at 13% PAA and 0.33% bis-acrylamide cross-linker. Polymerization was initiated by addition of 2 μL of tetramethylenediamine and 4 μL of ammonium persulfate.

Before loading the samples into the torsion rheometer, test samples were cut into circular disks. We recommend 5- to 10-mm tall and approximately 20 mm in diameter. A cork borer or tissue punch works well to punch a circular disc from a sample (Fig 5a).

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The soft samples tested here adhered well to the plates of the torsion rheometer, but solids that are too firm may slip against the plates and will not work as well. We also note that this design does not work well for Newtonian fluids or Maxwell viscoelastic fluids, which will creep without retaining their shape between the plates.

C. Work flow for torsion rheometer

1. Performing the test

The sample to be measured should be placed on a flat surface below the upper rotational plate of the torsion rheometer. The user then lowers the upper rotational plate to make contact with the sample. Care should be taken to align the center of the upper plate with the center of the sample. For best results, the entire surface of the sample should be in contact with the upper plate (Fig 5b). The position of the upper plate with respect to the sample can be readjusted by lowering and raising on the stand as needed. A permanent marker can be useful for marking the center of placement for the sample on the top and bottom plates.

Once the sample is in contact with the upper plate, data collection is started, zeroing the sensor at its current position. The test is then initiated by twisting the top plate, which applies a small initial angular displacement. This twist should be made very quickly and should be small enough that the sample does not slip, usually through an angle of 30 degrees or less. For soft viscoelastic solids, the applied displacement causes the plate to oscillate back and forth over time. The amplitude of the oscillations decays over time at a rate proportional to the viscous damping of the sample. The resulting motion of the plate is measured by the rotary sensor, which reads the angle of the plate, $\theta$, over time. Sample data are shown in Figure 6. The data are well described by damped harmonic motion. Typically, 2 to 20 peaks can be observed before the oscillations completely die out, with more oscillations for stiffer, less damped materials. If you cannot observe 2 peaks, apply a twist with a larger amplitude, as a minimum of 2 peaks is needed for the measurement.

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The readout from the test is used to calculate G′ and G″ through 2 main parameters: (a) the angular frequency $\omega$ of the oscillations in rad/s and (b) the ratio of amplitudes between subsequent peaks. The frequency of the oscillations is given by $\omega =2\pi /T$, where T is the time interval between subsequent peaks. The rate of amplitude decay is given by the ratio of the amplitude of the first peak and the amplitude of the second peak, ${\theta }_1/{\theta }_2$. If the motion is strongly damped, an alternative calculation can use the time $t_m$ and the amplitude ${\theta }_m$ of the first minimum, where T is given by $2\left(t_m-t_1\right)$ and the ratio ${\theta }_1/{\theta }_2$ is given by ${\left({\theta }_1/{\theta }_m\right)}^2$. These data can be read directly off the graph by selecting points in the LabQuest display or read by the user from a data table provided by the Vernier device or software. With LabQuest, the data can also be saved to analyze later.

2. Calculating the moment of inertia of the torsion plate

The moment of inertia I of the rotating plate is given by $I=\frac{1}{2}m{r}^2$, where m is the mass of the plate and r is its radius. If the top plate has more than 1 assembly piece, then the total moment of inertia is given by the sum of each piece, with each point mass m adding $I=m{d}^2$ for d, the distance of the mass from the axis of rotation.

3. Calculating sample shape factor

The strain on a sample depends not only on how it is displaced but also its size and shape. To account for these variations, we defined the sample shape factor s as (8) $$s=\frac{\pi R^4}{2h}$$ where R is the radius of the sample and h is the height of the sample.

4. Calculating shear and loss modulus

The mechanical properties of the test samples can be calculated based on the results of the torsion test. The shear modulus G′ and the loss modulus G″ can be computed from the frequency ($\omega$) and the logarithmic decrement of the amplitude $\Delta$, defined here as $\Delta =\text{In}\left({\theta }_1/{\theta }_2\right)$. The logarithmic decrement ($\Delta$) is a metric of the amplitude decay quantifying how damped the system is. The shear modulus G′ and loss modulus G″ are then given by (9) $$G\prime =({\omega }^{2}I/s)(1+{\Delta }^2/4{\pi }^2)$$ and (10) $$G\prime =({\omega }^{2}I/s)({\Delta }^{\pi })$$ where I is the moment of inertia of the rotating plate and s is the shape factor (Eq. 8), a quantity measured in units of volume. This can also be used to show that G′ and G″ are measured in units of pressure, Pa or kPa.

We provide a full derivation of these equations from the experimental variables T, ${\theta }_1$, and ${\theta }_2$ in the Supplemental Material, in which the Kelvin–Voigt model parameters $\kappa$ and $\beta$ appear as intermediate quantities.

IV. RESULTS AND DISCUSSION

Table 2 shows the average shear modulus G′ and loss modulus G″ for our 4 experimental samples. Also in Table 2, we compare the values of G′ and G″ obtained using the benchtop torsion device to results obtained using a commercially available rheometer. Overall, we find strong agreement between the benchtop torsion and commercial rheometer for a range of shear storage moduli G′ ranging from 0.8 to 30 kPa. We note that there are discrepancies in the experimental data between the torsion rheometer and the commercial rheometer. This is likely caused by the limitations of controlling strain and frequency in the benchtop torsion rheometer, which cannot be as sensitively controlled compared to the commercial rheometer. In the benchtop rheometer, the strain is dependent on how hard the user pushes the plate, whereas the frequency is set by the moment of inertia of the top plate and viscoelastic properties of the sample. Nonetheless, the data presented in Table 2 are close to the values obtained with the commercial rheometer. These results indicate that the torsion rheometer can be reliably used to obtain viscoelastic data of complex, viscoelastic solids.

Table 2 Results from the classroom rheometer design as well as the laboratory rheometer. G′ and G″ were determined on a commercial Malvern Kinexus rheometer with an oscillatory shear test at 2% strain and frequency 1 rad/s. Error denotes standard deviation of 3 independent samples.

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In many materials, the shear modulus G′ and storage modulus G″ depend on the frequency of oscillation. For our torsion rheometer, the frequency of oscillations is an emergent dependent parameter of the torsion rheometer's design, being a complex function of the material properties, sample shape, and moment of inertia of the top plate (Supplemental Material). Stiffer material increases the torque on the plate and the restoring force, increasing the frequency. Likewise, a larger sample radius increases torque on the plate and increases the frequency. In contrast, the moment of inertia of the top plate is inversely related to the frequency: a smaller top plate is easier to move, thus having a higher frequency than a large top plate with a larger moment of inertia.

Here, a systematic way to vary the frequency of oscillation is to alter the moment of inertia of the top plate. Figure 7 shows such a design. A simple outrigger (e.g., a ruler) is added to the top plate, and weights are added to the ends of the outrigger to alter the moment of inertia. More mass increases the moment of inertia of the upper plate and decreases the frequency of oscillations.

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Figure 7 shows frequency sweeps using our torsion rheometer for 3 different samples: tofu, agar, and a linear PAA gel. The range of frequencies we obtain from this design with these samples is 0.5 to 5 Hz. As shown in the data, tofu and agar exhibit a frequency-dependent result in which G′ increases with oscillation frequency. In contrast, the PAA sample exhibits a nearly constant G′. These trends are consistent with results obtained from a commercial rheometer (Fig 7c). The range of frequencies of the torsion rheometer and its manual nature are much more limited compared to the commercial rheometer, which can easily apply frequencies from 0.1 to 10 to the sample via a force transducer. Nonetheless, the torsion rheometer is able to reliably vary oscillation frequencies and detect frequency-dependent changes in shear modulus for such materials.

We note that our analysis (Eqs. 9 and 10) has assumed that shear and loss modulus of the sample is independent of the strain amplitude or frequency at which the material is probed. This is not always a justified assumption. Commercial rheometers face the same challenge for highly nonlinear complex materials (3, 6) for which G′ and G″ are not strictly defined. The storage and loss modulus reported by a commercial rheometer is often the best-fit value from the collected stress–strain data over the course of an oscillation. An alternative approach often used for nonlinear materials and for large-amplitude oscillations is to analyze the stress–strain curves directly and choose the appropriate model (6, 7). One limitation of the basic design we have presented is that the torsion rheometer does not have a torque sensor to directly obtain the stress–strain data over the course of an oscillation. Given the modular nature of the torsion device, a force transducer could be incorporated for studies of nonlinear materials.

B. Example experiments

To improve student learning and allow student discovery of material properties and experimental design, we propose a series of simple tests and experiments.

1. Frictional calibration

One limitation of the torsion rheometer device is friction of the rotary plate. The friction of the rotatory plate itself can be computed by spinning the rotatory device in the absence of a material sample and then allowing it to rotate until it slowly comes to a stop due to friction. This takes several minutes, indicating a very small coefficient of friction. Students can be asked to measure the friction and interpret the limitations of the device in measuring G′ and G″. For the setup used here, we computed a frictional torque of 6.2 × 10⁻⁵ N m.

2. Measuring axial normal force

In addition to oscillatory shear tests, the benchtop rheometer can be easily modified to measure axial normal force and apply uniaxial compression to samples. The axial normal force can be measured in newtons (N) by swapping out the bottom plate with a scale such that the force can be read off by converting the displayed mass to kilograms and multiplying by g, where g is acceleration due to gravity, 9.81 m/s² (Fig 8). Supplemental Movie S1 shows a torsion rheometer test in which oscillations in the normal force (∼1 N) can be detected as the plate oscillates back and forth in shear.

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Uniaxial compression can be applied to samples by manually lowering the top plate down via the stand or by placing a laboratory jack underneath the sample and raising it up toward the top plate. The axial compressive strain of the sample can be determined by measuring the gap height d with a ruler. The axial strain is then given by d/h × 100%, where h is the initial height of the sample. The value of G′ and G″ can then be determined as described above for different values of axial strain.

Figure 8 shows the shear modulus G′ of tofu and agar samples. For both samples, as they are compressed, G′ increases, indicating that the samples are stiffer compared to their uncompressed state. This is a feature common to many viscoelastic solids and biological materials, such as cells and tissues, and is called “compression stiffening” (11–13). This test can be used to connect concepts from physics and engineering to biology and medicine, as the onset of compressive stresses and dysregulation of physical stresses is associated with many different types of cancer (14–16) and tissue stiffness can be used as an early diagnostic marker for diseases, such as liver disease (17, 18).

3. Bacterial growth on substrates of varying mechanical properties

In recent years, the field of mechanobiology has emerged to study the effects of mechanical forces on biological systems (19–21). It is increasingly recognized, for example, that the mechanical properties of a cell's environment can affect many cellular functions, including motility, differentiation, and proliferation (22–24). An example of this is the growth of bacterial colonies on agar substrates of varying stiffness. Most bacterial colonies tend to grow slower and smaller on more concentrated, stiffer agar (25, 26). A series of agar substrates of varying concentrations (0.4% to 2%) can be prepared and characterized by the torsion rheometer (Fig 9). Students can experiment with different types of bacteria, such as E. coli or other bacteria that they can isolate from soil (27, 28). Students can inoculate bacterial colonies on agar plates and then observe and document the colony growth. The growth rate can be computed by measuring the average size of colonies over multiple days. Students can note changes in colony morphology and use smartphones to document the colony growth (Fig 9).

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V. CONCLUSIONS AND STUDENT FEEDBACK

We have demonstrated the feasibility of a low-cost, effective, and portable way to measure the mechanical properties of complex, viscoelastic materials. This tool can be used to incorporate viscoelastic materials and biophysical concepts into high school–level classes. The skills obtained in these laboratory activities provide students with a foundation that can better prepare students for their college course work and future lab experiences.

Students' feedback on what part of our outreach lesson stood out to them often mentioned seeing and interpreting the rheometer output in real time. Overall, the rheometer was a very useful tool in showing how to quantify the intuitive properties of elasticity and viscosity. Using the rheometer stimulated discussion among the students and helped make concrete the concepts that were discussed. Some reflections from student feedback include the following:

“I learned about scientific examples made out of everyday things.”

“It made me think of applications of elastic/viscous materials as shock absorbers.”

“I loved this lesson. I feel like I have a really great grasp on the concept now.”

One teacher mentioned the demo sparking “conversations with students about the applications of viscoelasticity in other fields [food and engineering]. I will be sure to include related examples in the future.” In an earth science classroom, 1 teacher remarked, “This gave me some great ideas on teaching fluid dynamics under the Earth's crust.”

Overall, the classroom rheometer proved to be an engaging teaching device for students in biology, physics, and earth science.