IPhO 2018, Експеримент — Задача 2. Viscoelasticity of a polymer thread

Part A: Stress-relaxation measurements (1,9 т.)

Please note that the thread must not be stressed before the beginning of the experiment! In case the thread is inadvertently stressed, ask for a spare one, but be reminded that this will take some time, therefore reducing the time you have for your experiment.

You should read carefully the indications given on "Part D: Data Analysis" before starting the measurements on this part in order to plan the way you make the measurements.

A.1 Measure the length of the unstretched thread between the screw heads. To obtain the total thread length, $\ell_0$, including the length inside the screws, add 5 mm for each screw. Write down in the answer sheet the measured value of $\ell_0$ and its uncertainty. [0,3 т.]

A.2 Measure the total weight of the mass-set, $P_0$, in gram-force (gf) units. Remember that 1 gram-force is the force corresponding to the weight of a mass of 1 gram ($1\ \mathrm{gf} = 9.80 \times 10^{-3}$ N). Write down in the answer sheet the measured value and an estimation of its uncertainty. [0,3 т.]

To observe experimentally the various relaxation components it is necessary to measure the stress for a long enough time. In this case, it is sufficient to sample the stress evolution during about 45 minutes.

You should now perform two simultaneous actions 1. and 2. Please read the instructions carefully before starting.

Important: if the experiment is interrupted for any reason it cannot be resumed. It has to be restarted with a new thread. In such case, ask for a spare one.

Take the following simultaneous actions:

1. Keeping the mass-set on the scale platform, stretch the thread so that the holding screw at the opposite side is placed on the thread supporting system, at the standing structure (Figure 4).
2. Start the chronometer simultaneously with action 1.

A.3 Record the readings of the scale, $P(t)$, and the corresponding reading instant, $t$, during around 45 min, in the table provided in the answer sheet. [1 т.]

A.4 Measure the length of the stretched thread, $\ell$, and estimate the corresponding uncertainty. Write down in the answer sheet the measured value of $\ell$ and its uncertainty. [0,3 т.]

Part B: Measurement of the stretched thread diameter (1,5 т.)

Never look directly at the laser! When not in use, the laser pointer should be off.
If you have difficulties in getting a diffraction pattern, please ask for a new laser.

In this part you will use light diffraction to measure the diameter of the polymer thread. The nominal diameter of the unstretched thread is $0.5$ mm. As you may know, the diffraction pattern of a rectangular slit of width $d$ is similar to that of a cylindrical object with the same diameter $d$ as the slit width. In the far-field (Fraunhofer) regime, where the diffraction pattern is observed in a target screen placed at a distance much larger than the diameter of the object, the distance between the diffraction minima for small angles is the same for both the slit and the object and is given by

where $\theta$ is the diffraction angle.

You will use laser light with a wavelength $\lambda$ = $650 \pm 10$ nm.

To perform this part, proceed as follows:

1. Turn on the laser using the spring clamp (see Figure 5).
2. Position the laser so that it hits the stretched thread directly.
3. With the provided material, devise a method to project the diffraction pattern into a paper screen, and to measure the data needed to determine the diameter of the thread using eq. (10).

B.1 Make a sketch of your method in the answer sheet. [0,6 т.]

B.2 Measure the optical distance, $D$, between the thread and the projected diffraction pattern. Write it down in the answer sheet with an estimation of its uncertainty. [0,3 т.]

B.3 Determine the average distance, $\bar{x}$, between diffraction minima and its uncertainty. Write it down in the answer sheet with an estimation of its uncertainty. [0,3 т.]

B.4 Applying eq. (10) to your diffraction data, determine the diameter, $d$, of the stretched polymer thread and its uncertainty. Write it down in the answer sheet with an estimation of its uncertainty. [0,3 т.]

Part C: Changing to a new thread (0,3 т.)

Before proceeding with the data analysis (Part D) you have to prepare the setup for the measurement with the shorter thread (Part E).

Detach the mass-set from the long thread (unscrewing it) and transfer it to the free end of the shorter thread (inserting the thread through the hole and fixing it with the screw-thread, see Figure 6).

In case you are unable to insert the thread through the hole, please ask for help.

C.1 Measure the length, $\ell_0'$ of the thread as in A.1. Write it down in the answer sheet with an estimation of its uncertainty. [0,3 т.]

Hang this new thread on the upper support so that the mass will exert a constant stress. The thread will eventually reach the stationary strain $\epsilon = \sigma/E$, while you work out the data analysis (it should be suspended for at least 30 minutes).

Part D: Data analysis (5,7 т.)

N.B.: The acceleration of gravity in Lisbon is $g = 9.80\,\mathrm{m\,s^{-2}}$ .

D.1 Calculate the force on the thread, $F$, in gf, for all data points and fill the corresponding column in the table used in A.3. [0,3 т.]

D.2 Plot $F(t)$ in the graph paper provided in the answer sheet. [0,4 т.]

Since the scale platform does not move, the measurements can be considered at constant strain and eq. (9) can be used. The ratio $\frac{\sigma}{\epsilon}$ can be written as $\frac{\sigma}{\epsilon} = \beta F$, where $\beta$ is a constant. Therefore,

where the sum was ordered $(\tau_1 > \tau_2 > \tau_3 > ...)$ for convenience.

D.3 Determine the constant strain, $\epsilon$, and the corresponding uncertainty. Write it down in the answer sheet with an estimation of its uncertainty. [0,3 т.]

D.4 Calculate the factor $\beta$, with $\sigma$ in SI units and $F$ in gf units. Write it down in the answer sheet (no uncertainty required). [0,3 т.]

D.5 Look at the data in the graph used in D.2: it cannot be explained by a purely elastic process. Sketch qualitatively in the graph paper provided in the answer sheet what you would expect for $F(t)$ in the purely elastic case. [0,4 т.]

The data analysis is easier if we consider $\frac{\mathrm{d}F}{\mathrm{d}t}$ instead of $F(t)$. This means that the relaxation parameters can then be extracted by hand in successive steps. In order to do this, the time derivative $\frac{\mathrm{d}F}{\mathrm{d}t}$ should be calculated at every point. This can be done either graphically or numerically. In the simpler case where the data points are taken at equal intervals, the numerical value of the derivative of a function $f(t)$ at point $t_i$, in a data set $(t_1, f_1), (t_2, f_2), (t_3, f_3), \cdots$, is approximately given by

where $h$ is the (constant) interval between the points and $N$ is the number of points.

If the intervals between data points are not equal, the numerical value of the derivative is approximately given by:

where $h_+ = (t_{i+1} - t_i)$ and $h_- = (t_i - t_{i-1})$ and $N$ is the number of data points. This expression represents the average derivative at left and right, weighted by the inverse time interval.

To analyse the data and extract the relevant parameters it is necessary to follow a sequence of steps. Hence, given the ordered sum in equation (11), do the following:

D.6 Assume that your data set lasts for longer than $\tau_2$ and calculate the derivative, $\frac{\mathrm{d}F}{\mathrm{d}t}$ , for data points at times $t > 1000$ s. Register the values in the table used in A.3. In case you use a graphical method for calculating $\frac{\mathrm{d}F}{\mathrm{d}t}$, use the graph paper provided in the answer sheet. [0,5 т.]

D.7 In the answer sheet, write an expression for the expected time dependence of $\frac{\mathrm{d}F}{\mathrm{d}t}$ in the case of a single viscoelastic process. [0,3 т.]

D.8 Extract, using a graphical method, the parameters $E_1$ and $\tau_1$ in SI units from the data points referred in D.6. Write $E_1$ and $\tau_1$ in the answer sheet (no uncertainties required). [1 т.]

D.9 Extract the parameter $E_0$ in SI units from the data points referred in D.6. Write it down in the answer sheet (no uncertainty required). [0,3 т.]

D.10 Fill column $y(t)$ in the table used in A.3 by subtracting the elastic and the longest viscoelastic components from $F(t)$ (the points used in D.6 do not need to be considered here). [0,3 т.]

D.11 Extract from $y(t)$ (see D.10), using a graphical method, the parameters for the second viscoelastic component, $E_2$ and $\tau_2$, in SI units. Write $E_2$ and $\tau_2$ in the answer sheet (no uncertainties required). [1 т.]

Additional viscoelastic components can be extracted in a similar way.

D.12 Identify the time window $[t_\mathrm{i}, t_\mathrm{f}]$ relevant for the third component. Write $t_\mathrm{i}$ and $t_\mathrm{f}$ in the answer sheet (no uncertainties required). [0,3 т.]

D.13 Estimate $\tau_3$ in SI units from the graph in D.11. Write it down in the answer sheet (no uncertainty required). [0,3 т.]

Part E: Measuring $E$ in constant stress conditions (0,6 т.)

Go back to the shorter thread suspended in Part C. Make sure that at least 30 minutes have passed since the thread was suspended. You can now safely assume that this thread has reached the stationary value of the strain $\epsilon = \sigma/E$.

E.1 Determine $E$ directly from the length of the stretched thread. Write it down in the answer sheet, together with the relative difference to the value $E_0$ obtained in Part D (no uncertainties required). [0,6 т.]