A. J. Etzei, S. J. GoldsIein, H. J. Panabaker, D. G. Fradkln, and L. H. Sperling Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015 Rubber is composed of long polymer chains. While natural rubber is made from cis-polyisoprene, synthetic rubber is often made from statistical copolymers of butadiene and styrene. T o suppress flow of one chain past another, the polymer chains are cross-linked, a process sometimes known as vulcanization. The process of vulcanization transforms the linear chains into a three-dimensional network. The product is known as an elastomer. In the relaxed state, the polymer chains of an elastomer form random coils. On extension. the chains are stretched out, and their conformational entropy is reduced. It is this reduced entrow that makes rubber hands "snar, back" when released. his bhenomenon is identical to the entropy increase obtained when releasing a gas from under pressure. The decrease in entropy on &etching an elastomer is exDressed analytically. by. the statistical theory of rubber elas&city (1.2). Stretching a rubber band makes a good classroom demonstration of the stress-strain behavior of cross-linked elastomers. This paper descrihes such an experiment, comparing the statistical theory of rubber elasticity with its continuum mechanics counterpart. The former follows the equation of state of ruhher elasticitv. (I. . . 2 .) and in the latter the MoonevRivlin equation ( 3 , 4 )is employed. Theory The theory of rubber elasticity (1,2) explains the relationship between stress and strain in terms of the number of active network chains and temperature. An active network chain is defined as a portion of a chain bound on both ends by crosslinks. For ordinary rubber, there may be 5-10 active network chains per primary chain. The equation of state for rubber elasticity may be written o = nRT ( a
- l/a2)
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Flgwe 1. Simple rubberelastic behavior of a rubber band under increasing load. Note nonlinear behavior.
Expanding the Mooney-Rivlin equation to include a third term, the following equation results (6): Equation 1 yields the temperature and crosslink level dependence of the stress-strain relationship, with no arbitrary constants. Equations 2 and 3 yield increasingly closer fits of the data, but with arbitrary constants determined from the experiment.
(1)
where o = stress, Pa; n = number of active chain segments per unit volume in network, motlm3; R = gas constant, 8.314 X Jlmol K; T = absolute temperature of experiment, K; and a = extension ratio LILn. . ",dimensionless. The strain is (L Lo)/Lo, or a - 1. Equation 1shows that the stress-strain is non-Hookian: i.e.. stress is not s i m ~ l ~ v r o ~ o r t i o nto a lstrain. The quantity n is calculated from i h e iktial slope of the ex~erimentalstress-strain curve. Eauation 1also expresses the correct temperature dependence. The Moonev-Rivlin equation expresses the relationship between stress-and strain-containing the elastomer as a continuum where 2C1 and 2Czareconstants. The interpretation of these constants has been the subject of much study; however, the results are inconclusive. According to Flory (5),the ratio of 2C212Cl is related to the looseness with which the crosslinks are embedded within the structure, while others cite nonequilibrium effects. According to eq 1, the quantity ol(n - l/a2) should be constant, but eq 2 predicts that this quantity depends on llar. A plot of o/(a - I/&) versus l/a will yield the constants 2C>and 2Co from the intercent and slone. . . res~ectivelv. . ~'ollowingthe work of ~ o i n emore ~ , generalized tieories of the stress-strain relationships in elastomers were sought.
Experiment Time: About 30 min Level: Introductory Polymer Science or Physical Chemistry Principles Illustrated: behavior of an elastomer on extension rubber elasticity theory characteristicsof an elastomer as described by the Maoney-Rivlin equation Equipment and Supplies: 1large ruhher hand 1set of weights, up to 25 kg 1meter stick 1wire coat hanger bent so weights can be attached 2 hooks of >I14 in. diameter to hold rubber hand first, measure and record the length of the rubber band in the relaxed stare. Weigh the hanger and hook to he placed st the hottom of thp ruhher hand so that the total force on the ruhher hand ran he calculated. Suspend the rubber band from a high place using the second hook. Beein bv,.olacine a weieht on the haneer and measure ,, thplangrhafther!rhher hand Record the total werghtsandmeasure thenrrrpaponding rubber hand lengthuntiltheruhher hand l,reak3. Use caution when nearing this point Results and Discussion
A plot of stress (using initial cross sectional area) as a function of a , Figure 1, demonstrates the nonlinearity of the stress-strain relationship. The sharp upturn of the experiVolume 63 Number 8 August 1986
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Figure 2.) These values are in general agreement with literature values (8).The data can also he fitted to eq 3 and the constants, C, C', and C", can be determined. Equation 3, with two additional terms over the statistical theory of rubber elasticity fits the upswing a t low 11a values in Figure 2. All numerical results are summarized below: A = 5.04 X
m2 (cross sect. area)
2C, = 2.29 X 105Pa
Figure 2. Mwney-Rivlin plot d the data presented in Figure I . The upturn a t high extension is due to narrGaussian behavior arm is accounted tor by the C" term.
mental data a t a strain of four is due to the limited extensibility of the chains themselves ( 1 , 6 ) . The number of active network chains per unit volume, n, was found to be 1.94 X lo2molIm3. This value was compared with that determined from a swelling experiment (7) on a similar rubber band, 1.55 X 10%mol/m3. These twovaluesare in agreement and illustrate that the fundamental quantity n may be determined more than one way. (For systems where the stoichiometry is known, the quantity n may also be calculated from the crosslinkine chemistrv.) A plot according to the ~ o o n & ~ i v l i equation, n Figure 2, vields a curve that ra~idlvdecreasesfor values of l/ameater than 0.25. The cons&nti2& and 2C2 from eq 2 werecalculated from the intercept and slope, respectively, of the linear portion of the curve. (The lowest a point has some experimental error, and was ignored in drawing the straight line in
732
Journal of Chemical Education
By way of summary, important chemical properties of elastomers, such as the number of active network chains per unit volume, can be rapidly and easily estimated by simple extension using the theory of rubber elasticity. The stressstrain curve mav he accuratelv fitted bv the Moonev-Rivlin equation or by forms containing higher-terms nt high extensions. This demonstration ex~erirntmtteaches the interrelationships among molecular characteristics, mechanical behavior, and modern polymer science. Literature Cited 11) Billrneyer,F. W.. Jr. "TertbookofPolymer Seiena"; 3rded.. Wiley-Interscience:Nev
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(21 Mark, J. E. JChem.Educ. 1981,58,898. (3) Mmnev,M. J.Appl.Phva. 1948,19,134. IB Rivlin, R. S. Philoa. Trow. R0.y. Soe. London, A 1948,240,159,491,~. (51 Flory, P. J. Proe. Roy. Soc. Lond0n.A 1918,351,351. 161 Shen, M. In "Science end Teehnologv of Rubber"; Eirieh, F. R.. Ed.: Academic: New York, 1978. (71 Sperling, L. H.; Michael, T. C. J. Cham. Educ. I982,59,651. (8) Nwrderrneer. J. W. N.; Ferry, J. P. J. Polym. Sci.,Polym. Phys. Ed. 1976,14,5W.
