Rubber Elasticity A Simple Method for Measurement of Thermodynamic Properties John P. Byrne University of Technology, Sydney, Australia, 2007
Elastomers constitute an important class of polymeric materials. Their structure, mechanical behavior, and thermodynamic properties have been the subject of a number of articles (1-8)in this Journal. Perhaps their most striking feature is their thermal behavior. Elastomers, when heated under load, contract rather than expand. Experiments illustrating qualitative aspects of this phenomenon have been described by Clough (4)and Bader (5);however, elastomers also provide an interesting alternative system for the measurement of quantitative thermodynamicfunctions. The thermodynamics of rubber elasticity has been discussed by Nash (7) and others (6, 8)in this Journal, and now the topic is included in some recent physical chemistry texts (9,lO). Acomprehensive treatment of the subject is given in the text by Treloar (11). If rubber is stretched elastically by a force f, then the differential of the work done by a further small extension, dl, is given by dw = f a - P d V (1) For most rubbers, the extension can be considered to occur at constant volume, so that thePdV work term is negligible. For a system at constant volume and temperature, the differential work term, dw, is equal to the change in Helmholtz free energy, so for the elastomer dA=dw=fdl
or
(6A/6DT= f
(2)
(3)
The Helmholtz free energy of the rubber can be expressed in terms of its internal energy U and entropy S, as A=U-TS
(4)
and the variation of A with length, a t constant temperature, can be derived as (6A/8OT= (6U161)~ - !lY6SlS1)~
(5)
Using eq 4, the restoring force in stretched rubber can then be expressed as f = (6U/&l)T-T(&S161)T or
f=f,+f.
Figure 1. Adapted triple-beam balance used for measuring the force applied to the rubber sample.
(6)
(7)
Thus, the restoring force in stretched rubber can be interpreted in terms of two thermodynamic components, f, arising from changes in internal energy with length, and f., that arises from changes in entropy with length. In this experiment, the rubber is held at constant length, or strain, and heated in a water bath. As the temperature increases, the force or load required to maintain constant length will increase. According to eq 6, a plot of force against temperature should yield a straight line whose slope gives (~SISI)T for the rubber, and whose intercept is (SUIGI)T. The apparatus can be constructed easily and cheaply from a modified triple-beam balance. Anovel feature is the use of an optical lever to detect small changes in the length of the rubber.
Experimental The apparatus consists of an adapted Ohaus triple-beam balance, mounted on a suitable timber support (Fig. 1). The balance pan was removed to allow space for the thermometer. The mass of the pan was compensated for by a suitable counterweight added to the pan support. The rubber sample was stretched between two brass hooks, one hung from the bottom of the balance pan support (A), and the other an adiustable lower hook (B). Holes were drilled in the timber frame, at sultahle locations, to accommodate the thermomerer and the two brass hooks. The lower adjustable hook passed through a brass collar attached to the timber frame, and could be tightened by means of a thumb screw. The initial elongation of the stretched rubber sample was controlled by means of the adjustable lower hook, and variations in load on the rubber of up to 500 g were controlled by varying the masses on the balance beams. The rubber was heated, under load, by means of a water bath, consistingof a l-L beaker of water heated by a 400-W laboratory hotplate stirrer. The temperature of the stirred water bath was measured using a 0-110 OC thermometer. Since the experiment requires constant volume conditions, i. e., measurements at constant length or strain, a suitably sensitive strain gauge is required. As only small deviations in elongation can be measured directly from the balance arm pointer, a light lever was used to amplify this movement. This was constructed by mounting a concave mirror (2-cm diameter, focal length around 35 cm) on the end of the balance arm (C). The light source was provided by a Volume 71 Number 6 June 1994
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quartz-halogen bulb (12 V, 55 W),mounted in a square section metal housing in which a horizontal slit (5 mm x 1 mm) was cut. The light source was mounted adjacent to a vertical scale constructed from a 10-cm length of millimeter scale ruler. These were supported by a retort stand and placed on the laboratory bench a t about 70-cm distance from the mirror. The mechanical advantage of this optical lever is about 15:l. The minimum detectable deviation of the light pointer, of around 1mm, corresponds to a change in length of 70p (or about 0.07% for a 10-cm length of stretched rubber). This precision is comparable with that obtained from the triple-beam balance (0.2 g in 350 g or 0.06%). The high precision of the final results hinges on the use of the light pointer that will detect changes in length not achievableby a more conventional direct needle pointer. The rubber sam~lesused were commereiallv available elastic bands of mass 0.3 g, with a 2.5 x 1 mm ckss section and unstretched lenah ofabout 50 mm. It is essential that the bands exhibit elistic behavior during the experiment, and no plastic deformation or creep occurs. Because creep is dependent on both elongationand temperature, the bands must be "condirioned before use. This was achieved by stretching the bands to an elongation greater than that used in the experiment (around 150%), and heating the bands to a temperature above that used in the experiment. The bands were stretched on a bent wire frame, immersed in a beaker of water, heated to 80°C, and left immersed for five minutes. If this precaution is not taken, creep will occur during the experiment, giving nonlinear results. After preconditioning, the band is mounted between the two hooks in the apparatus and the lower hook adjusted to give an extension of at least 100%. The band is then immersed in the water bath, at mom temperature, and the masses on the balance arm adjusted until the light pointer is just below (about 1cm) the arbitrary zero on the strain eauee. The load reauired is de~endenton the dimensions band, and thkextension.'ln our experiments loads of around 350 rr were twical. When the band has tbermallv equilibratedythe heat& is switched on. As the temperature rises, the rubber contracts, and the light pointer moves up the scale. When the pointer reaches the zero mark on the scale, the temperature is recorded, along with the load, and then an additional 5 g is added to the balance arm. Heating is then continued until the pointer again rises to the zero mark on the strain gauge, when the temperature is again noted. Repeated measurements of load and temperature, all at the same constant strain reading, are taken at 5-g load intervals. Measurements are discontinued at the first sign of creep. Even though the rubber has been preconditioned, creep may occur at temperatures above 65 "C. Results and Discussion
This experiment has been part of our second-year laboratory course in thermodynamics for the past few years. About 30 students per year complete the experiment, and the results obtained are surprisingly good considering the simplicity of the apparatus. Atypical plot of restoring force versus temperature is shown in Figure 2. The restoring force (N)was obtained bv converting .. the balance arm mass to kilograms and multiplying by acceleration due to gravity, 9.806 ms Loads range from 335-380 g, and the temperature increase for a 5-g load increment is typically 5 4 ' C . The slope and intercept of tho straight line in Fihwre 2 were calculated using a linear reh~essioncomputer program. Values of the thermodynamic functions.. ~ n- eq. - v e bv 6, are then obtained from,
'.
*
(6S/61),= -slope = 48.80 0.04)x lo3 N K' 532
Journal of Chemical Education
Temperature (K) Figure 2.Variation of restoring force with temperature for an elastic band. and The correlation coefficient (12) is 0.9998. The two highest temperature points show deviation from linear behavior, with the onset of creep, and were not included in the regression analysis. Values off, and f, can then evaluated as follows: f, = (6U/6DT=0.69+ 0.01N
The value of the entropic contribution to the restoring force,f., can be calculated at room temperature, T = 298 K, from fa = -'Z'(&SIS1)T= 2.62 + 0.01 N Students can relate these measured thermodynamic functions to the properties and structure of rubber by discussing the following points. The experimental value for the change in entropy with extension, (6S181)~= -8.8 x lo3 N K-', is negative. This decrease in entropy with lemth can be related to the incmased order thai ;esults as tile coiled polymer chains are alimed and ordered when the elastomer is stretched. The restoring force in stretched rubber can be interpreted in terms of the two measured thermodynamic functions, f, and f, (eq 7). The results show that the force arising from changes in entropy with length (f, = 2.62 N) is three to four times larger than the contribution from changes in internal energy with length (f, = 0.69 N). This illustrates the important role of entropy in elastomer behavior. The stretching of an elastomer pmvides an illustrative thermodynamic analogue for the compression of a gas. (SUI81)Tfor the elastomer is equivalent to (8U16V)~ for the gas. For the isothermal compression of an ideal gas, The nonzero value of (6U/&, measured in this experiment, can be related to nonideal behavior in gases, and discussed in terms of deviation of the rubber from "ideal" behavior. Finally, the deviation from linearity of Figure 2, at high temperatures, introduces the concept of plastic deformation or creep. The experiment can be extended by repeating the measurements with a nonconditioned baud. In this
case, creep is much more dramatic and begins at lower temperatures. Acknowledgment
I am grateful to Hans Gotthard for his assistance with the design and construction of the apparatus. Literature Cited 1. Kauffmann, G. B.; Sqmmu, R. B. J Chrm. Educ. 1890.67.422425,
2. Fad"wez, F:Mathias, L. J.: Kmaehiultz,J.;Carraher, C. E,J. Chem. Educ. 1988, 65. 352355. 3. Q X ~ I J. ~ IP; . M& J. E J them. E ~ U C1987, 64, 4914%. 4. C1ough.S. B . J . Chom. Educ. 1 9 8 7 , 6 4 , 4 ~ . 5. B ~ ~Ms J~ them. , E ~ W 1981.58.285. . 6 . Mandelkem, L J Chrm. Educ. 1918.55.177-181. I. N S S ~ , LK. . J them. E ~ X C ls~s,56,36%368. . 8. he=. E.; K L.; J. il them. E ~ U C .l s n , s o . 753.756. 9. Alberty, R. A.: Silbey. R. J. P h y s i d Chamistry; W~ley:New York, 1992: p 135. 10. A t a s . P W. Phydml Chrmisfni: 0.U.P:Oxford. 1990; 4th ed., p 713. 11. Trelosr L R. G. ThoPhysi.s$RubholElaslicity,3rd ed; Clarendm Ress: Oxford. 1975.
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