Elastomers

Abstract

In this group research project you will investigate some of the fascinating properties of rubber both experimentally

and computationally. You will examine the thermodynamics of ideal rubber and devise a simple 1D computational

model of the polymer chains. Ideal rubber is, of course, a mathematical abstraction and so you will also investigate the

expansion of rubber in the laboratory and use this to decide whether or not real rubber can be treated as ’ideal’. You

will investigate the expansion of rubber as a function of both load and temperature. Typically, the thermal expansion

coefficient of unloaded, non-ideal, rubber is small and, in this experiment, you will build a Michelson interferometer

to accurately measure the small changes in length. You will document your findings as a report written in the style

of a scientific paper from the APS journal Physical Review B.

1 Introduction

Rubber is an elastomer, or ’elastic polymer’. It is an incredibly versatile material that is used in manyapplications; from car tyres, to running shoes, to the pencil ’rubbers’ that first gave this material its name.The properties of elastomers are remarkable and very different to almost any other solids. For example,ideal rubber does not expand when heated. In fact, loaded rubber will actually contract if heat is applied.Rubber is a particular type of elastic polymer formed by vulcanising latex, a natural extract from the rubbertree. After vulcanisation (a procedure invented in 1839 by Charles Goodyear, of Goodyear tyre fame) thelong polymers of natural rubber are interlinked by sulphur compounds. Here we will describe the rubber interms of the polymer chains between these sulphur crosslinks: rubber is essentially a 3D zigzag structurewith freely rotating bonds that allow the polymer chains to change their length by coiling or uncoiling.

In most solids, the atoms or molecules are held in place by strong intermolecular potentials. These determinethe equilibrium state of the solid and, because each atom or molecule sits in a deep potential well, it can take agreat deal of energy to deform or expand a normal solid. The thermodynamics of most solids are determinedby the variation of internal energy with shape and size and this also determines the thermal expansivity ofthe material. When heated most solids expand and this is because, with higher thermal energy, each atomor molecule can oscillate further from its equilibrium position leading to an effective increase in volume.

In ideal rubber on the other hand, each polymer chain is free to rotate about its bonds and each chain cantherefore coil or uncoil without changing the internal energy, U . The internal energy of rubber is thereforeindependent of the shape at constant volume and, like an ideal gas, rubber obeys Joule’s law: rubber canstretch and un-stretch with no change in U . However, we know that when rubber is stretched we feel arestoring force, so what is the origin of this force? The answer is that the force is entropic in nature.When rubber is stretched the polymer chains uncoil and begin to align into a more ordered state withcorrespondingly lower entropy. The force we feel is a direct result of the second law of thermodynamics:the rubber is trying to pull back into a more disordered state where its entropy is a maximum. Again, thisbehaviour is analogous (but opposite in sign) to the ideal gas.

1.1 Deliverables

To complete this group project you will need to research and understand some basic concepts ofThermodynamics including the first law of thermodynamics, and entropy. You will also need to learnabout optics and the Michelson interferometer. The project is split into two parts, an experimental partin which you will study the expansion of rubber in the laboratory and a theoretical/computational part inwhich you will investigate the properties and the behaviour of ideal rubber. You will report your findings ina single ’scientific paper’. This will be written in the style of the American physical society journal, PhysicalReview B [1]. Although this document contains a small amount of background it is essential that you study

many different sources to find all the information that you will need.

2 Theory: Thermodynamics of ideal rubber

The first law of thermodynamics is simply a restatement of the conservation of energy. It can be written as

Fdl = dU − TdS, (1)

where F is the force acting over distance dl, dU is the change in internal energy and dS is the change inentropy S at constant temperature, T . We will always work in quasistatic equilibrium, so the load F on therubber is always equal to the restoring force F generated by the rubber. Then,

F =

(

∂U

∂l

)

T

− T

(

∂S

∂L

)

T

. (2)

In Eq. 2 it is clear that there are two contributions to the load needed to stretch a substance, a contributionrelated to the change in internal energy with respect to length and a contribution from the change in entropywith respect to length. In most solids, the first term is large and the second negligible. In rubber (and theideal gas) the first term is negligible and it is the entropic term that is important.

The entropy is essentially a measure of the disorder of the system. For rubber, the entropy is large whenthe polymer chains are coiled and tangled, and small when the rubber is stretched so that the polymerchains uncoil and align. It is possible to obtain a good physical understanding of the entropy of rubber witha simple 1-dimensional model of a single rubber molecule [2]. In this model we assume that we have Nlinks of polymer each of length b and that these links are randomly ordered in 1D so that each link pointseither to the right or to the left. Clearly there are many different ways of arranging the links, and differentarrangements will give different lengths of polymer. This is identical to a random walk problem where youtake steps of length b randomly to the right or to the left and see how far you have travelled after N steps.

L

Figure 1: Left: 2D projection of a rubber polymer of 18 links, tangled in 3D. Right: 1D model of a rubber molecule.

The figure shows 1 possible arrangement of N = 8 links of length b, with total length l = 5b.

The entropy is related to the number of ways, W , in which each possible length of the rubber molecule canbe achieved. Each time the chains rearrange through random thermal motion each particular arrangementof links will arise with equal probability but some particular lengths of chain can be obtained in many moreways than others and so the random thermal motion will likely push the molecule into these most probablelengths. The Boltzmann relation (which is inscribed on Ludwig Boltzmann’s tombstone) relates the entropyto W ,

S = k lnW, (3)

where k is the Boltzmann constant.

In the random walk problem, the number of ways, W , of arranging N links with Nr links pointing to theright and Nl = 1 − Nr links pointing to the left is obtained simply from the binomial distribution,

W =N !

Nr!Nl!. (4)

As the length of the 1D rubber chain is related to the number of links pointing in each direction,

l = b(Nr − Nl), (5)

it is possible to relate the entropy to the length of the rubber chain. Then, from Eq. 2, the relation betweenforce, length and temperature can easily be deduced,

F = kTǫ/lo. (6)

where the stretch ǫ is defined by the ratio of the stretched length l′ to the equilibrium length lo, ǫ = l′/lo.In the 1D problem, although the most probable length of the rubber chain is zero, the average length isl0 = b

√N . Eq. 6 is the equation of state for ideal 1D rubber. It is analogous to the more familiar equation

of state for an ideal gas which relates the pressure, volume and temperature.

2.1 Tasks

• Investigate the random walk problem computationally. Devise a computational model tocalculate the length of a 1D rubber molecule after aligning its N links randomly to the right or the left.Investigate the distribution of lengths as a function of N and find the most probable configurations.What is the distribution function in the limit of large N? How does the shape of the distributionfunction change with N?

If you programme in C you may need to use the pseudo-random number generator from the standardlibrary. The following C-code fragment will print an integer pseudo-random number between 0 and thesystem variable RAND MAX.

#include <stdlib.h>

#include <stdio.h>

#include <time.h>

int main(){

srand(time(0)); /* initialise pseudo-random number generator */

printf("%d\n",rand()); /* random integer between 0 and RAND_MAX */

}

• Derive and investigate the equation of state for ideal rubber. By combining basicthermodynamics (Eq. 2) with your understanding of the random walk problem and the Boltzmannrelation you can derive the equation of state for rubber. You should investigate the equation of stateof ideal rubber and understand how the entropy, force, length and temperature are related.

Note: the one dimensional random walk model enables us to understand the majority of the physicsof ideal rubber, but in a full 3D model there are some small changes in the relation between entropyand length and hence in the equation of state. The Guth-James equation of state for 3D rubber [6] is

F =kT

lo(ǫ − 1/ǫ2). (7)

Stirling’s approximation, lnN ! = N lnN − N, may be useful.

3 Experiment: Expansion of loaded rubber

Eq. 6 predicts that the stretch, ǫ, of ideal rubber is directly proportional to the load and inverselyproportional to the temperature.

This can easily be tested experimentally by measuring the extension of a rubber strip as a function of eitherload, temperature, or both.

3.1 Tasks

• Investigate the expansion of rubber experimentally. Test the theoretical predictions you havemade and measure how real rubber behaves in relation to ideal rubber. What happens to real rubberat extremes of temperature or load?

You will need to think very carefully about the design of your experiment and how to minimise errors.You will have access to a rubber strip, water bath, clamps, pulleys, and retort stands, a travellingmicroscope, a ruler, thermometer, newton meter and a selection of weights.

4 Experiment: Thermal expansion coefficient of unloaded rubber

The thermal expansion coefficient is defined as,

α =1

L

dL

dT. (8)

For most materials this is small, Tipler [3] gives values between 10−6 to 10−5 K−1 for various materials fromDiamond to Copper. For non-ideal real rubber the quoted values vary widely, for example from 2.2 × 10−4

K−1 [4] to 7.7 × 10−5 K−1 [5]. Pellicer et al [6] derive the following result by considering the equation ofstate and the 1st law,

(

∂U

∂L

)

T

= AT 2α

(

ǫ +2

ǫ2

)

, (9)

where A is a positive constant and α = 1

Lo

dLo

dTis the thermal expansion coefficient of un-stretched rubber

[6]. If the rubber is un-stretched, then from Eq. 9 we can write,

α =1

3kT 2

(

∂U

∂L

)

T

, (10)

where α is usually assumed to be constant and the sign of α clearly depends on the sign of(

∂U

∂L

)

T. Ideal

rubber is a Joule’s’ law substance ((

∂U

∂L

)

T= 0) so the deviation of the thermal expansion coefficient from

zero will give us some information about how ’ideal’ real rubber actually is.

Conceptually, measuring the thermal expansion coefficient of a material is easy, we simply need to measurethe length as a function of temperature. In practice, however, this can be difficult. Essentially we needto measure changes in length of the order of just a few 100 nanometres, and one of the few ways of doingthis accurately is to use a laser interferometer. Figure 2 shows a schematic of the experimental set-up that

Figure 2: The experimental set-up.

you will use to measure the thermal expansion coefficient of rubber. The key components are those onthe optical bench. The laser (wavelength λ), beam splitter and the mirrors M3 and M4 form a Michelsoninterferometer. The purpose of Mirrors M1 and M2 is simply to adjust the height of the beam coming fromthe laser. Depending on the optical components available they may not be needed.

The operation of the Michelson interferometer is relatively simple and is explained in detail in Ref. [7].Imagine that the optical path lengths B → M3 and B → M4 are the same, then the light travelling fromB to M3 and back to B will travel exactly the same distance as the light going from B to M4 and back toB. These light waves will interfere constructively at B and a bright central spot will be seen on the screen.If the movable mirror M4 is moved by one quarter of a wavelength each wave arriving at B will be halfa wavelength out of phase, and the central spot will change from bright to dark. If the movable mirror isthen moved by a further one quarter of a wavelength, the central spot will again be bright and each fringewill have moved to the position previously occupied by the adjacent fringe. So, by counting the number offringes n that move past a fixed reference point we can accurately determine the distance ∆L moved by themirror,

∆L = nλ

2. (11)

Figure 3: Example output from the photodiode: 20 bright fringes counted in 400 seconds corresponding to a

temperature change of 4.2 K.

In Fig. 2 the heat chamber can be considered as a ’black box’ that contains the mirror, M4, mountedonto one end of a thin rubber bar of length 20 mm (diameter 10mm). The chamber also contains a heaterplate and two thermocouples, allowing the rubber to be heated and its temperature to be measured. Thethermocouple has been pre-calibrated between 0.1 and 0.5 V, and its temperature (◦C), voltage relation is,

T (V ) = −159.2 + 592.8V (12)

When the rubber is heated it may expand, moving the mirror M4 and changing the interference pattern.The number of fringes which pass a fixed reference point on the screen can be counted by eye, or by usingthe photodiode and pico-Logger software. The photodiode will register a high voltage if the fringe is brightand a low voltage if the fringe is dark (see Fig. 3). The lens is used to expand out the interference pattern sothat it can be viewed on a screen at a reasonable distance from the beam splitter (note that the beam splitterin the lab contains a compensator plate). The purpose of the mirrors M1 and M2 is to give a horizontalbeam aligned at the height of the rest of the optics in the experiment. This experiment can be challenging.It is essential to align the optics correctly and some time (and patience) should be expended optimising thealignment.

4.1 Tasks

• Set up the experiment as suggested in Fig. 2 and measure the expansion coefficient of

rubber as a function of temperature. What does the value you measure tell you about the ’ideal-ness’ of real rubber. How much does the expansion coefficient vary as a function of temperature andtheoretically how might you expect it to vary?

As usual you should analyse and discuss the sources of error and limitations in the experiment andthink about how to minimise these.

The following hints may help when constructing a Michelson interferometer.

1. Set up the laser with the beam parallel to the surface of the optical bench.

2. Adjust the heights of M1, M2, and BS so that the beam hits the center of each when you slide eachinto the beam. Mark the center of the screen E at the same height.

3. Set the optical path BS to M1 equal to BS to M2 within 1 mm or so if possible

4. Cover M2. Adjust orientation of BS so that it reflects the beam returning from M1 into the centre ofthe screen.

5. Uncover M2. Note the appearance of an additional bright spot on the screen. Again, ignore the weakspots.

6. Adjust the orientation of M2 so that the brightest spot on the screen due to the beam from M2 preciselycoincides with the brightest spot from M1. Interference often causes the brightness of the combinedspot to flicker.

7. Insert a 40x microscope objective lens (L3) between BS and the screen to make the fringes visible.

References

[1] http://prb.aps.org, see papers in the current issue and instructions for authors.

[2] I. Muller and P. Strehlow, Rubber and Rubber Balloons: Paradigms of Thermodynamics, Springer (2004).

[3] P. A. Tipler, Physics for scientists and engineers, 4th Ed., W. H. Freeman and Company, p634.

[4] Polymer Handbook, edited by J. Bandru, E. H. Immergut, Wiley New York (1975).

[5] http://www.engineeringtoolbox.com/linear-expansion-coefficients-d 95.html

[6] J. Pellicer, J. A. Manzanares, J. Zuniga and P. Utrillas, J. Chem. Ed 78, 263, (2001).

[7] E. Hecht, Optics, 2nd Ed., Addison-Wesley, sect. 9.4.2.

Mervyn Roy, 2011

S.A Atarah

Using the ADC 11 picoLog Recorder

S. A. Atarah (2008)

The following are guiding steps to setting up and using the software which controls

the pico data logger to collect data. Use the following steps to set up the software.

Login to Windows

Click Start>All Programs >Pico Technology>PicoLog Recorder On the ‘Welcome to PicoLog for Windows…’ screen, choose ‘Normal’

On the main window, go to File>New Settings Choose the following settings;

Recording method Real time continuous Action at end of run Stop Restart delay 1 minute

Use multiple converters No The figure illustrate each stage of the configuration process.

Press ‘OK’

Choose the following settings;

Sampling interval 1s Maximum number of samples 1500 (or other suitable) Readings per sample As many as possible

S.A Atarah

Press ‘OK’

Choose the following

Converter type ADC-11 (USB) USB Devices:

Device ADC-11 (USB) Serial KJLxxxx

Press ‘OK’. This opens the window entitled ‘ADC11/22 measurements’ Press ‘Add’. This opens up the window entitled ‘Edit ADC11 measurement’ You may enter a short text for Name to describe the channel or leave as blank. In the

Channel field, enter a digit corresponding to the channel number or use default

number. Press ‘OK’. One channel has been set up.

Press ‘Add’ again and repeat previous step till enough channels have been set up.

S.A Atarah

Go to File>New Data Enter a file name for the data to be collected, then press ‘OK’. Further settings such as data resolution can be made by clicking on Options … at the stage. To start collecting data press the red ‘Start recording’ button.

Once started, the data may be viewed as a sheet or a graph. Click on the button with a

picture of (a grid) a graph on it. This will present a real time (sheet) graph of the data.

Left clicking will zoom in and right clicking will zoom out. In the top left hand corner

there is readout of the coordinates of the crosshairs. So, hovering over a point of

interest on the graph will give you the time and voltage coordinates.