Leonard K. Nash Harvard University Cambridge, Massachusetts 02138
Entropy and Rubbery Elasticity
Rubbery elasticity demands the presence of long polymeric molecules.
The statistical theory of rubbery elasticity confronts a characteristically chemical system that has a fascination all its own. hut the mathematical demands imnosed hv conventional &positions of the theory have long iestricte-d its presentation. A chaotic tangle of nolvmeric chain segments does indeed appear a formidible sibject for analysis,~articularly when compared with the apparent simplicity of the ideal gas. Of the two, however, statistical analysis of the ideal rubber is actually much the easier, both mathematically and conceptually. Obstructed by no enigmas of indistinguishability and quantization, the analysis can proceed as a straightforward exercise in Boltzmanu statistics, calling for no mathematics beyond simple algebra and the bare idea of a derivative. Such an exnosition of the theorv fits well into the verv first nresrntaiion of itatistical ~ h e r m u d ~ n a m i and c s hah'proved i l l tellieihle cvrn tu wlleee freshmen with nl, stutis~icalbackbeyond, say, ccapter 1of (1) Naah. L. K.. "ChemThermo." Addison-Wesley, Reading, MA, 1372.
General Sources Fortunate is the teacher who has access to the recent revision of what has long been the definitive monograph (21 Trelaar, L. R G., "The P h p i i i ,f Ribbebe - l E
1975. Rather more widely distributed is a classic treatment that places ruhberv elasticitv in the general context of macromoiecular behavior
Abundant journal references may be found in recent reviews like
Teachers and advanced students will profit from an exposition of James' and Guth's sophisticated statistical theory, as given by (61 M4uarrie. D. M.."StafiaieslThormodynemics."Harper & Row. New York, l913,ch. 14.
and also in older statistical thermodvnamics texts bv D. ter Haar and A. H. Wilson. A simpler theory, further simplified in the following exnosition. has recentlv been restated hv its original autho; 171 Wa1l.F. T.. "ChomicalThemodynamica." 3rd ed., Freeman. San Francisco, 1974,ch.
16. An older statistical thermodynamics book by T. L. Hill also presents this theory which, among large modern physical chemistry texts, seems to appear only (with utmost brevity) in
(81 Adamson, A. W.."Terthk ofphysical Chemistry."AeademiePress, New York, 1973, pp. 1021-1025.
Even beginning students can understand the limited exposition of the same theory given in a paperback by (91 Treloar. L. R. G.. "Introduction ta Palmer bienee." Springer Verlag, New York. 1970.
131 Flory, P. J.."PrinciplesofPolperChemistry."Corndl University Press, lthaca NY, 1953. chapters 10 & 11.
The is one of a series of Resource Papers, intended primarily for college and university teachers. The publication of the series is supported in part by a grant from the Research Corporation. Leonard K. Nash received the PhD degree in Chemistry from Harvard University in 1944. After a year with the Manhattan Project and a year teaching at the University of Illinois, he returned to Harvard, where he holds the rank of Professor of Chemistry and served as Chairman of the K' Nash Chemistry Department during the years 1971-74. He received the Manufacturing Chemists Teaching Award in 1966,the James Flack Norris Award in 1975, was for five years a member uf the Advisory Council on College Chemistry, and has served on NSF delegations to Japan and India. The author of many technical hooks and articles, Professor Nash has also written several hooks on the history and philosophy of science. He is a member of Phi Beta Kappa, Sigma Xi, and the American Academy of Arts and Sciences.
and, in rather less detail, in another paperback by I101 Msndelkcm, Lco."An Introduction to Mamomolecules," Sprinppr Verlag, New York. 1972. See also his recent article in J. CHEM EDUC., 55. 177 ll9781.
Thermodynamic Analysls Observing how very little volume change accompanies the isothermal stretching of rubber, we simplify our analysis by supposing d V r-. 0. More specifically, consider that when elastic behavior comes in question the thermodvnamic work 6u8 = -1'dV + /dl., where? symtx,li?es a uniaxik stretching tijrce and dL the rlistance through which it acts. r'l'he diffcrt'nc(! in sign arises, Iwause whert.posirive d l ' reflects a work ourpur, -pusitivc dL n.flerts a wnrk input). UDt o 4everd at. mospheres pressure, IPdVl _< 0.001 l i d ~ intheisothermal I stretching of rubber, so the P d V component of work may here be ignored without significant error. In place of the familiar equation d E = T d S - PdV, the operative relation now becomes ~
dE = TdS + fdL Consider next what happens when, freed from all restraint, a moderately stretched piece of rubber snaps back to its relaxed length. That snapback produces, we find, virtually no Volume 56. Number 6. June 1979 / 363
...chain segments lie in relatively compact random coils, like a great tangle o f "molecular spaghetti."
change of temperature in either the rubber or its surroundings. Such uniformity of temperature implies virtually zero heat transfer in a free contraction yielding zero work output. With 69 0 = Sw, the isothermal free contraction then necessarily proceeds with d E 0. But, energy being a function of state, it follows that even in the presence of beat and work transfers, and for extensions iust as for contractions. in real rubber all isothermal alterations of length proceed w i h dE 0. And so we come to define a n ideal rubber as one that suffers no enerw -. change in isothermal extension or contraction. The statistical theorv of rubherv elasticitv stands independent of the assumption d E = 0;but this akumption enormouslv-simolifies . the develo~mentof such a theorv. bv" reducing our last equation to
-
-
-
f = -T(dSldL)? (1) where the subscript underlines the vital constant-temperature condition. And having now paralleled a familiar argument that takes departure from the Joule expansion, we may conjecture that. even as in ideal -gas.. so too in ideal rubber the total entropy is made up of fully separable configurational and thermal components. In that event, ( d S 1 d L ) ~will represent nothing but a change in configurational entropy potentially assignable by statistical analysis of an appropriate model.
The Model Rubbery elasticity demands the presence of long polymeric molecules abundantly endowed with easily rotatable links. 1) Natural rubber is cis-1.4-polyisoprene,with an average molecule comprising some 5000 isoprene units joined by C-C single bonds
about which rotation is, we suppose, relatively unhindered. 2) In X-ray diffraction pictures of unstressed rubber we find only the few diffuse rings characteristic of nearest neighbor distances in a liouid lackine . ohase . " anv . lone-ranee .. .. order. Interactions brturen nrighhoriny rmderulrsare then weak and i,rnndirrc. tiowl rwuglb to have rutatiunal freedom ,ubsrantially unim-
paired.
3) Full elastic recoverydemands that, even under sustained stress, individual polyisoprene molecules shall suffer no irreversible
net displacements. Forging sulfur bridges that tie these molecules into a single network, "vulcanization" adequately ensures the dimensional stability of rubber when no more than 1-2% of the isoprene monomers are thus cross-linked, and so denied their rotational freedom. Terminating at the crosslinks will be molecular segments comorising on averaEe some 100 isoprene units. When unstressed, these chainsegments lie in relatively compact random coils, like a great tangle of "molecular spaghetti." But, the 100-odd monomer units being free to rotate with respect to one another, the chain segments are free to assume very much more extended conformations. A readilv intellieible origin for the characteristically enormous extensibility of rubber is here found in the reserves of length supplied by compact coils. Observe too that even in radical changes of conformation no chemical bonds need be hent, stretched, or broken; and that, in the quasi-liquid system, the forces between partially straightened chain segments cannot much differ from the forces between highly convoluted chain segments. Conformational change is then attended by little alteration in either energy or volume, and our model thus duly 0 dV we associate with the evokes the conditions dE isothermal deformation of real rubber. If changes of conformation proceed through relatively unhindered internal rotations, and with net ( d E l d L ) ~= 0, why then is stretching a t all resisted? Resistance arises simply
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364 /
Journal of Chemical Education
because the oartial straiehtenine of chain seements is an ordering procek reflectedyn an e&opy decrease. Exactly like the isothermal compression of ideal gas, the isothermal stretching of ideal rubber is then a nonspontaneous change that demands a work input and delivers a heat output. Contrariwise, when released from stress, the chain segments' instant relapse into the chaotic disorder of random coils will he reflected in an entropy increase that makes the snapback of stretched rubber just as intelligibly spontaneous as the free expansion of compressed gas. The Conformatlonal Multiplicity of Polymeric Molecules The elasticallv active chain seements constitute the "molecules" whose configurational entropy is now to be quantified hy statistical analysis. Temporarily ignoring the angular constraints that limit the flexibility of real polymer chains, we begin with the statistically more tractable ideal polymer chain made up of uniform links so freely jointed that any link can even double back on its immediate predecessor.
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Freely-Jointed Chain in One Dimension
Let the free end of the first link lie at x = 0, and let the conformation of the one-dimensional chain be determined bv successive tosses of a coin. If a first toss of the coin turns up heads, we lay in the +x direction the chain's first link which, with length 1, then extends from x = 0 to x = +l. If a second toss of the coin also turns up heads we lay the second link from x = +1 on to x = +21; hut if the second toss turns up tails we will lay the second link of the chain in the -x direction, from x = +1 back to x = 0. Thus interpreted, n coin tosses will define one of the many conformations open to a chain of n links. Now we see at once that a long chain is most unlikely to lie fully extended. If the far end of the chain is to fall at x = +nl, the n tosses of a well-balanced coin must all turn up heads. With n = 100, there are 21Wpossible outcomes for the series of coin tosses, of which only one represents the unvarying seauence of 100 consecutive heads. And we well know that any long series of tosses of a well-balanced coin is most likely to yield heads and tails in comparable numbers-simply because such an outcome can arise in not just one but myriad possible ways. By the very same token, our polymeric chain is most likelv to contain com~arablenumbers of links oriented in the T X and - x directions and, in contrast 10 full extension, such hiehlv folded chain confirurntions are stronelv iavored simolv beiaise they can arise in not just one h u t myriad possibie wavs. Preciselv how stronelv - . favored is discoverable bv a simple exercise in Boltzmann statistics. If a one-dimensional chain begins a t the origin and ends at some x = ml, the integer rn will represent the margin by which the number of links oriented in the + x direction exceeds the number oriented in the -I direction. The number of ways (W,) in which a chain of n links can come to terminate a t x = ml is then necessarily equal to the number of ways in which a sequence of n coin tosses can yield a number of heads (H) that exceeds the number of tails (T) by the margin m. Just like chain links, each of the coin tosses is rendered unequivocally distineuishable bv in a sequence. Thus we mav -its olacement . be sure that W , = n!lH!T!
But with H+T=n
and
H-T-rn
A readily intelligible origin for the characteristically enormous extensibility of iubber i s found in the reserves of length supplied b y compact coils.. .
it follows that
H=-n + m
T = -n1
and
2
-m
2
'
substitution of which produces
.
.
For large n this function is easily evaluated by calling on Stirling's approximation in the short form: In y! = y In y y. This gives n+m n +m In W , = Inn! ---In 2
2
+n+m I n-m 2
2
--
n-m n - m -+ 2 2
Factoring and separating the In terns, we obtain after routine cancellations and rearrangements
Disregarding the improbable configurations approaching full extension. we confine our attention to those states in which m 1, and A, = A, < 1; but eqn. (12) clearly applies also to the uniaxial compression where, with A, < 1, A, = A* > 1duly reflects the consequent increase in cross-section. At this point the remaining thermodynamic analysis rapidly attains its goal. With u still denoting terms referring to an initial unstressed state, the change in configurational entropy
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A complementary differential coefficient arises from our L,/L,,. At a definition of the fractional deformation A, given temperature L,, is a constant characteristic of the unstressed ruhher, so that ( d h / d L , ) ~= 1/LZu
For the force directed along the x-axis, eqn. (1)assumes the form: f, = - T ( d s / d L , ) ~ . This the chain rule permits us to recast as fz
= -T(dSldh)~(dA,/dL,)~
Substitution for the two differential coefficients then produces NkT
1
Let us divide both sides of this equation hv the cross-sectional area (A,) of the unstressed bar. Much as we use pressure to represent force/area, we can (and. in work with elastomers. often do) speak of a nominal stresso I flA,. And in the de: nominator on the right the product AWL,, is nothing but the invariant total volume ( V )of the rubber. With these substitutions, and deletion of the now-superfluous subscript, the last equation becomes
which is significantly reminiscent of the ideal gas law. The quotient N/V represents the number of elastically active chain segments in unit volume of rubber. If by u we symbolize the number ofmoles ofchain segments per unit volume, then N/V = uN, where N stands f~ Avogadro's number. Since the ideal-gas-law constant R = Nk, our equaVolume 56,Number 6, June 1979 1 387
Considering the extreme simplicity of our theory, even such order-of-magnitude agreement represents a genuine triumph. Experiments and Problems for Students
The range of relevant experiments is limited primarily by the ingenuity of students and their teachers. Our short list of examples begins with three papers that deal with the design and the (Carnot) analysis of rotary heat engines in which rubber is the working substance. (11) Arehibald,P.B.,Sei. American. 224,118(Aprii1971). (121 Mullen, J. 0 . . and eoworkers,Am. J.Phys., 43,349 (1975); 46,1107 (1978).
For a novel engine of this type operating a t over 1000 rpm, consult (13) Fanis, R. J.. P d y m En& S c i , 17,737 (1977).
A
representative o versus h plot tor compression and elongation ot rubber.
tion of state for rubber can be rewritten in final form as
A splendid first approach to partial derivatives is opened by ex~erimentaldetermination of the familiar cvclic ~roduct.as . . cited by (141 Carmll, H. B., Einer, M., and Hanson, R. M., Am. J.Phyh.. 31,397 (1963).
Attractive experiments on the inflation of balloons are described by Experimental Jests
Redrawn from pp. 87 and 90 of reference (21, the figure disnlavs a renresentative u versus A nlot for rubber suhiected to h i a x i a l c&mpression (A < 1) andelongation (A > lj. (For purely practical reasons, the data on uniaxial longitudinal compression were actually derived from experiments on the equivalent biaxial latitudinal stretching). When the constant u is assigned an optimal magnitude, our equation produces a (aolidl t,heoretical line which handsomelv matches the distinctive shape of the (dashed) empirical stress-strain curve UD t o A 6. And the sham diveraence in the hiah-A - region offers no occasion for concern, since we know in advance that our simple Gaussian expressions must fail whenever chains make even a distant approach to full extension-beginning a t about 300% elongation, or A 4. Thus it is only in the region 1.4 < A < 4 that theoretical and experimental curves manifest significant ouantitatiue divergence. If the whole of the divergence cannot be dismissed as due to failure to achieve eauilihrium in the stress-strain measurements, that will onlv s&fy the need for some more elaborntr rhrory thisdispense< with crudear~r~roximnti~~ns hke dE ? O dV. Such theones continue to be much studied. With u as the only adjustable constant, eqn. (14) creditably accommodates empirical data on uniaxial compression and elongation. Now, far from being freely adjustable, u would be fully defined if we knew enough about the elastomeric network created by vulcanization. In a comparatiue sense, the values of u that best fit stress-strain measurements are duly observed t o rise in direct proportion to the measurable extent of crosslinking. But detailed information about the elastomeric network is hard to come by, and particularly problematic is the proper allowance for physical entanglements that create elastically active chain segments over and above those horn of croaslinking. Quantitatiuely, values of v estimated from the extent of crosslinkingdo, as expected, fall short of those values that best fit stress-strain measurements, but in order of magnitude the two sets of values are in good agreement.
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~
~
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368 1 Journal of Chemical Education
(151 Stein, R. S.,J. CHEM EDUC., 35,203 (1958). (161 Candon, F.E., and Fwd, M., J. CHEM EDUC., 35.518 (1958).
and with less experimental but more theoretical detail, by (171 Merritt, 0. R., and Weinhaus, F.. Am. J.Phya., 46,876 (19781.
Interesting problems here arise even from numerical plug-ins, e.g., establishing an average molecular weight for the chain segments. More ambitious problems for homework (or longer projects) are likely to fall under three main heads. First: derivations from eqn. (13) of other manifestations of rubbery elasticity, e.g., the relation given in reference (15) for balloon inflation. Second: other applications of Bernoulli statistics to phenomena of diffusion, heterogeneous equilibrium, etc. A few of these are treated with some rigor hut little mathematics in (181 Villan, F. M. H..and Benedek, G. M., "Statistical Physic$," constituting vol. 1 of "Phyaicp aith Illustrative Examples fmm Medicine and Biolagy," Addison-Wesley, Readine MA. 1974.
and an extremely simple measurement of diffusion rate is used in an attractive determination of Avogadro's number eiven (19) A1erandrakis.G.C.. Am. J. Phys.. 46.810(1978).
Third: classical thermodynamics of elastic systems, like Gough-Joule effects and other striking relations of partial differential coefficients. Apart from references (2) and (3), useful material may be found in (20) Hill,T.L.,"ThermdynarninforChernis~andBiologi~~,"Addiaan-W~ley,Reading. MA. 1968. eh. 3.
and at an elementary level in (21) Bent, H. A , "The Second Law," Oxford U.P.. New York,1965,eh. 37,
The thermodynamic concept of "availability" displays its power in an application to the eauilihrium of interconnected balloons reported by (22) Weinhsua, F., and Barker, W. Am. J.Phys.. 46.978 (1978).
